NTCIR12-MathWiki-1rate: 0

i = 0 n - 1 i 0 normal-… n 1 i=0\ldots n-1

NTCIR12-MathWiki-1rate: 0

( - n k ) = - n - ( n + 1 ) - ( n + k - 2 ) - ( n + k - 1 ) k ! = ( - 1 ) k n ( n + 1 ) ( n + 2 ) ( n + k - 1 ) k ! = ( - 1 ) k ( n + k - 1 k ) = ( - 1 ) k ( ( n k ) ) . binomial n k absent fragments n normal-⋅ fragments normal-( n 1 normal-) normal-… fragments normal-( n k 2 normal-) normal-⋅ fragments normal-( n k 1 normal-) k missing-subexpression absent superscript 1 k normal-⋅ n n 1 n 2 normal-⋯ n k 1 k missing-subexpression absent superscript 1 k binomial n k 1 k missing-subexpression absent superscript 1 k binomial n k \begin{aligned}\displaystyle{\left({{-n}\atop{k}}\right)}&\displaystyle=\frac{% -n\cdot-(n+1)\dots-(n+k-2)\cdot-(n+k-1)}{k!}\\ &\displaystyle=(-1)^{k}\;\frac{n\cdot(n+1)\cdot(n+2)\cdots(n+k-1)}{k!}\\ &\displaystyle=(-1)^{k}{\left({{n+k-1}\atop{k}}\right)}\\ &\displaystyle=(-1)^{k}\left(\!\!{\left({{n}\atop{k}}\right)}\!\!\right)\;.% \end{aligned}

NTCIR12-MathWiki-1rate: 1

j 2 A ( - 58 2 ) = 396 4 , e π 58 396 4 - 104.00000017 formulae-sequence subscript j 2 A 58 2 superscript 396 4 superscript e π 58 superscript 396 4 104.00000017 normal-… j_{2A}\Big(\tfrac{\sqrt{-58}}{2}\Big)=396^{4},\qquad\quad e^{\pi\sqrt{58}}% \approx 396^{4}-104.00000017\dots

NTCIR12-MathWiki-1rate: 0

i = 0 n - 2. i 0 normal-… n 2. i=0\ldots n-2.

NTCIR12-MathWiki-1rate: 0

Total resources available = Resources at start of innings - Resources lost by first interruption - Resources lost by second interruption - Resources lost by third interruption - etc Total resources available Resources at start of innings Resources lost by first interruption Resources lost by second interruption Resources lost by third interruption etc normal-… \begin{matrix}\,\text{Total}\\ \,\text{resources}\\ \,\text{available}\end{matrix}\ \ \ =\ \ \ \begin{matrix}\,\text{Resources}\\ \,\text{at start}\\ \,\text{of innings}\end{matrix}\ \ \ \ -\ \ \ \ \ \ \ \ \ \ \ \ \ \begin{% matrix}\,\text{Resources lost by}\\ \,\text{first interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ % \ \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{second interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ % \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{third interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ \ % \ \,\text{etc}...

NTCIR12-MathWiki-1rate: 0

{ 0 , 1 p - 1 } 0 1 normal-… p 1 \{0,1\dots p-1\}

NTCIR12-MathWiki-1rate: 0

2 x - 2 x - y 2 x 2 x y \displaystyle 2x-2x-y

NTCIR12-MathWiki-1rate: 3

ψ = 1 - 5 2 = 1 - φ = - 1 φ - 0.61803 39887 ψ 1 5 2 1 φ 1 φ 0.61803 39887 normal-⋯ \psi=\frac{1-\sqrt{5}}{2}=1-\varphi=-{1\over\varphi}\approx-0.61803\,39887\cdots

NTCIR12-MathWiki-1rate: 0

ϵ n = ϵ - K p . . ( 16 ) fragments subscript ϵ normal-n ϵ subscript K normal-p normal-… normal-… normal-… normal-. normal-. fragments normal-( 16 normal-) \;\epsilon_{\mathrm{n}}=\epsilon-K_{\mathrm{p}}...........(16)

NTCIR12-MathWiki-1rate: 0

2 π - 2 φ 2 π 2 φ 2π-2φ

NTCIR12-MathWiki-1rate: 2

τ 5 = β - α 60 - 0.01667 α + 0.01667 β subscript τ 5 β α superscript 60 0.01667 α 0.01667 β \tau_{5}=\frac{\beta-\alpha}{60^{\circ}}\approx-0.01667\alpha+0.01667\beta

NTCIR12-MathWiki-1rate: 0

4 t - 2 n 4 t 2 n 4t-2n

NTCIR12-MathWiki-1rate: 2

φ = 1 - 5 2 = - 0.6180 339887 φ 1 5 2 0.6180 339887 normal-… \varphi=\frac{1-\sqrt{5}}{2}=-0.6180\,339887\dots

NTCIR12-MathWiki-1rate: 1

e π 19 ( 5 x ) 3 - 6.000 010 e π 43 ( 5 x ) 3 - 6.000 000 010 e π 67 ( 5 x ) 3 - 6.000 000 000 061 e π 163 ( 5 x ) 3 - 6.000 000 000 000 000 034 superscript e π 19 absent superscript 5 x 3 6.000 010 normal-… superscript e π 43 absent superscript 5 x 3 6.000 000 010 normal-… superscript e π 67 absent superscript 5 x 3 6.000 000 000 061 normal-… superscript e π 163 absent superscript 5 x 3 6.000 000 000 000 000 034 normal-… \begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx(5x)^{3}-6.0% 00\,010\dots\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx(5x)^{3}-6.000\,000\,010% \dots\\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,0% 61\dots\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,% 000\,000\,034\dots\end{aligned}

NTCIR12-MathWiki-1rate: 0

H ( X 1 , , X n ) = - x 1 x n P ( x 1 , , x n ) log 2 [ P ( x 1 , , x n ) ] H subscript X 1 normal-… subscript X n subscript subscript x 1 normal-… subscript subscript x n P subscript x 1 normal-… subscript x n subscript 2 P subscript x 1 normal-… subscript x n H(X_{1},...,X_{n})=-\sum_{x_{1}}...\sum_{x_{n}}P(x_{1},...,x_{n})\log_{2}[P(x_% {1},...,x_{n})]\!

NTCIR12-MathWiki-1rate: 2

M = - 6 + 12 / 2 - 8 / 3 + 6 / 2 - 24 / 5 + = - 1.74756 . M 6 12 2 8 3 6 2 24 5 normal-⋯ 1.74756 normal-… M=-6+12/\sqrt{2}-8/\sqrt{3}+6/2-24/\sqrt{5}+\cdots=-1.74756\dots.

NTCIR12-MathWiki-1rate: 0

𝐦 = - γ 𝐋 𝐦 γ 𝐋 \mathbf{m}=-\gamma\mathbf{L}

NTCIR12-MathWiki-1rate: 2

( π + 20 ) i = - 0.9999999992 - i 0.000039 - 1 superscript π 20 i 0.9999999992 normal-… normal-⋅ i 0.000039 normal-… 1 (\pi+20)^{i}=-0.9999999992\ldots-i\cdot 0.000039\ldots\approx-1

NTCIR12-MathWiki-1rate: 2

ζ ( 2 ) = Derivative of ζ ( 2 ) = - n = 2 ln n n 2 = - 0.9375482543 superscript ζ normal-′ 2 = Derivative of ζ 2 superscript subscript n 2 n superscript n 2 0.9375482543 normal-… \scriptstyle\zeta^{\prime}(2)\,\,\text{= Derivative of }\zeta(2)=-\sum\limits_% {n=2}^{\infty}\frac{\ln n}{n^{2}}=-0.9375482543\ldots

NTCIR12-MathWiki-1rate: 0

{ 0 , i , i - 1 , - i , i - 1 , - i } 0 i i 1 i i 1 i normal-… \{0,i,i-1,-i,i-1,-i...\}\,

NTCIR12-MathWiki-1rate: 2

- 2 5 = - 1.148698354 5 2 1.148698354 normal-… \sqrt[5]{-2}\,=-1.148698354\ldots

NTCIR12-MathWiki-1rate: 2

ζ ( - 1 ) = 1 12 - ln A - 0.1654211437 superscript ζ normal-′ 1 1 12 A 0.1654211437 normal-… \zeta^{\prime}(-1)=\frac{1}{12}-\ln A\approx-0.1654211437\ldots

NTCIR12-MathWiki-1rate: 1

1 - 1 2 = - 0.41421 1 1 2 0.41421 normal-… 1-1\sqrt{2}=-0.41421\ldots

NTCIR12-MathWiki-1rate: 2

41 - 29 2 = - 0.01219 41 29 2 0.01219 normal-… 41-29\sqrt{2}=-0.01219\ldots

NTCIR12-MathWiki-1rate: 4

1393 - 985 2 = - 0.00035 1393 985 2 0.00035 normal-… 1393-985\sqrt{2}=-0.00035\ldots

NTCIR12-MathWiki-1rate: 0

2 n - h - 2 2 n h 2 2n-h-2

NTCIR12-MathWiki-1rate: 1

c 2 = - 1 6 ; c 3 = - 1 108 ; c 4 = 7 3240 ; c 5 = - 19 48600 fragments subscript c 2 1 6 normal-; subscript c 3 1 108 normal-; subscript c 4 7 3240 normal-; subscript c 5 19 48600 normal-… c_{2}=-\frac{1}{6}\quad;\quad c_{3}=-\frac{1}{108}\quad;\quad c_{4}=\frac{7}{3% 240}\quad;\quad c_{5}=-\frac{19}{48600}\ \dots

NTCIR12-MathWiki-1rate: 4

- 0.026838601 0.026838601 normal-… -0.026838601\ldots

NTCIR12-MathWiki-1rate: 0

2 - 2 g - n 2 2 g n 2-2g-n

NTCIR12-MathWiki-1rate: 2

x = - 1.84208 x 1.84208 normal-… x=-1.84208\dots

NTCIR12-MathWiki-1rate: 0

1 , 1 , 2 , 3 , 1 , - 2 , - 3 , - 5 , - 2 , - 3 , for the signs + , + , + , - , - , + , - , - , . 1 1 2 3 1 2 3 5 2 3 limit-from normal-… for the signs normal-… 1,1,2,3,1,-2,-3,-5,-2,-3,\ldots\,\text{ for the signs }+,+,+,-,-,+,-,-,\ldots.

NTCIR12-MathWiki-1rate: 2

θ ( 0 ) = - ln π + γ + π / 2 + 3 ln 2 2 = - 2.6860917 superscript θ normal-′ 0 π γ π 2 3 2 2 2.6860917 normal-… \theta^{\prime}(0)=-\frac{\ln\pi+\gamma+\pi/2+3\ln 2}{2}=-2.6860917\ldots

NTCIR12-MathWiki-1rate: 2

± 17.8455995405 plus-or-minus 17.8455995405 normal-… \pm 17.8455995405\ldots

NTCIR12-MathWiki-1rate: 0

e - δ τ superscript e δ τ e^{-\delta\tau}

NTCIR12-MathWiki-1rate: 0

2 a - R 2 a R 2a-R

NTCIR12-MathWiki-1rate: 2

lim k c 2 ( k ) = - 2.157782996659 . subscript normal-→ k subscript c 2 k 2.157782996659 normal-… \lim_{k\to\infty}c_{2}(k)=-2.157782996659\ldots.

NTCIR12-MathWiki-1rate: 2

e π 58 = 396 4 - 104.000000177 . superscript e π 58 superscript 396 4 104.000000177 normal-… e^{\pi\sqrt{58}}=396^{4}-104.000000177\dots.

NTCIR12-MathWiki-1rate: 0

2 - 2 g - n 2 2 g n 2-2g-n

NTCIR12-MathWiki-1rate: 2

γ 1 ( 1 2 ) = - 2 γ ln 2 - ln 2 2 + γ 1 = - 1.353459680 subscript γ 1 1 2 2 γ 2 superscript 2 2 subscript γ 1 1.353459680 normal-… \gamma_{1}\!\left(\!\frac{1}{\,2\,}\!\right)=-2\gamma\ln 2-\ln^{2}\!2+\gamma_{% 1}\,=\,-1.353459680\ldots

NTCIR12-MathWiki-1rate: 0

- Λ normal-Λ Planck-constant-over-2-pi -\Lambda\hbar

NTCIR12-MathWiki-1rate: 0

- ω δ ω δ -\omega\delta

NTCIR12-MathWiki-1rate: 2

z 1 = - 0.4121345 + i 0.5978119 subscript z 1 0.4121345 normal-… i 0.5978119 normal-… z_{1}=-0.4121345\ldots+i0.5978119\ldots

NTCIR12-MathWiki-1rate: 2

z 2 = - 1.4455692 + i 0.6992608 subscript z 2 1.4455692 normal-… i 0.6992608 normal-… z_{2}=-1.4455692\ldots+i0.6992608\ldots

NTCIR12-MathWiki-1rate: 0

2 π - ν 2 π ν 2\pi-\nu

NTCIR12-MathWiki-1rate: 0

ϕ 1 = - 30 + 30 subscript ϕ 1 superscript 30 normal-… superscript 30 \phi_{1}=-30^{\circ}...+30^{\circ}

NTCIR12-MathWiki-1rate: 0

1 - e - λ θ 1 - e - 2 π λ 1 superscript e λ θ 1 superscript e 2 π λ \frac{1-e^{-\lambda\theta}}{1-e^{-2\pi\lambda}}

NTCIR12-MathWiki-10rate: 0

k = 0 a k = k = 0 ( - 1 ) k ( k + 1 ) , superscript subscript k 0 subscript a k superscript subscript k 0 superscript 1 k k 1 \sum_{k=0}^{\infty}a_{k}=\sum_{k=0}^{\infty}(-1)^{k}(k+1),

NTCIR12-MathWiki-10rate: 1

ζ ( s , α ) = n = 0 1 ( n + α ) s = α 1 - s s - 1 + 1 2 α s + 2 0 sin ( s arctan t α ) ( α 2 + t 2 ) s 2 d t e 2 π t - 1 . ζ s α superscript subscript n 0 1 superscript n α s superscript α 1 s s 1 1 2 superscript α s 2 superscript subscript 0 s t α superscript superscript α 2 superscript t 2 s 2 d t superscript e 2 π t 1 \zeta(s,\alpha)=\sum_{n=0}^{\infty}\frac{1}{(n+\alpha)^{s}}=\frac{\alpha^{1-s}% }{s-1}+\frac{1}{2\alpha^{s}}+2\int_{0}^{\infty}\frac{\sin\left(s\arctan\frac{t% }{\alpha}\right)}{(\alpha^{2}+t^{2})^{\frac{s}{2}}}\frac{dt}{e^{2\pi t}-1}.

NTCIR12-MathWiki-10rate: 0

f ( z ) = n = 0 ( z - a ) n 1 2 π i C f ( w ) ( w - a ) n + 1 d w . f z superscript subscript n 0 superscript z a n 1 2 π i subscript C f w superscript w a n 1 normal-d w f(z)=\sum_{n=0}^{\infty}(z-a)^{n}{1\over 2\pi i}\int_{C}{f(w)\over(w-a)^{n+1}}% \,\mathrm{d}w.

NTCIR12-MathWiki-10rate: 1

S ν ( x ) = k = 0 sin ( ( 2 k + 1 ) π x ) ( 2 k + 1 ) ν subscript S ν x superscript subscript k 0 2 k 1 π x superscript 2 k 1 ν S_{\nu}(x)=\sum_{k=0}^{\infty}\frac{\sin((2k+1)\pi x)}{(2k+1)^{\nu}}

NTCIR12-MathWiki-10rate: 1

blanc ( x ) = n = 0 s ( 2 n x ) 2 n , blanc x superscript subscript n 0 s superscript 2 n x superscript 2 n {\rm blanc}(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}},

NTCIR12-MathWiki-10rate: 1

L ( s ) n = 0 ( - 1 ) n ( 2 n + 1 ) s L ( 1 - s ) = L ( s ) Γ ( s ) 2 s π - s sin π s 2 , formulae-sequence L s superscript subscript n 0 superscript 1 n superscript 2 n 1 s L 1 s L s normal-Γ s superscript 2 s superscript π s π s 2 L(s)\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}\qquad\qquad L(1-s)=L(% s)\Gamma(s)2^{s}\pi^{-s}\sin\frac{\pi s}{2},

NTCIR12-MathWiki-10rate: 2

Φ ( z , s , α ) = n = 0 z n ( n + α ) s . normal-Φ z s α superscript subscript n 0 superscript z n superscript n α s \Phi(z,s,\alpha)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+\alpha)^{s}}.

NTCIR12-MathWiki-10rate: 1

M ( a , b , z ) = n = 0 a ( n ) z n b ( n ) n ! = F 1 1 ( a ; b ; z ) , M a b z superscript subscript n 0 superscript a n superscript z n superscript b n n subscript subscript F 1 1 a b z M(a,b,z)=\sum_{n=0}^{\infty}\frac{a^{(n)}z^{n}}{b^{(n)}n!}={}_{1}F_{1}(a;b;z),

NTCIR12-MathWiki-10rate: 1

Z ( λ , ν ) = j = 0 λ j ( j ! ) ν . Z λ ν superscript subscript j 0 superscript λ j superscript j ν Z(\lambda,\nu)=\sum_{j=0}^{\infty}\frac{\lambda^{j}}{(j!)^{\nu}}.

NTCIR12-MathWiki-10rate: 0

n = 0 u n = n = 0 p ( n ) q ( n ) , superscript subscript n 0 subscript u n superscript subscript n 0 p n q n \sum_{n=0}^{\infty}u_{n}=\sum_{n=0}^{\infty}\frac{p(n)}{q(n)},

NTCIR12-MathWiki-10rate: 1

β ( s ) = n = 0 ( - 1 ) n ( 2 n + 1 ) s , β s superscript subscript n 0 superscript 1 n superscript 2 n 1 s \beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}},

NTCIR12-MathWiki-10rate: 1

L ( s , χ ) = n = 1 χ ( n ) n s L s χ superscript subscript n 1 χ n superscript n s L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}

NTCIR12-MathWiki-10rate: 1

L ( s , χ ) = n = 1 χ ( n ) n s . L s χ superscript subscript n 1 χ n superscript n s L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}.

NTCIR12-MathWiki-10rate: 1

ζ ( 2 s ) ζ ( s ) = n = 1 λ ( n ) n s . ζ 2 s ζ s superscript subscript n 1 λ n superscript n s \frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}.

NTCIR12-MathWiki-10rate: 1

F ( s ) = n = 1 f ( n ) n s F s superscript subscript n 1 f n superscript n s F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}

NTCIR12-MathWiki-10rate: 1

f [ x 0 , , x n ] = j = 0 n f ( x j ) q ( x j ) . f subscript x 0 normal-… subscript x n superscript subscript j 0 n f subscript x j superscript q normal-′ subscript x j f[x_{0},\dots,x_{n}]=\sum_{j=0}^{n}\frac{f(x_{j})}{q^{\prime}(x_{j})}.

NTCIR12-MathWiki-10rate: 1

f ( x ) = n = 0 s ( 2 n x ) 2 n f x superscript subscript n 0 s superscript 2 n x superscript 2 n \textstyle{f(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}}}

NTCIR12-MathWiki-10rate: 1

1 n = r = 1 1 ( n + 1 ) r . 1 n superscript subscript r 1 1 superscript n 1 r \frac{1}{n}=\sum_{r=1}^{\infty}\frac{1}{(n+1)^{r}}.

NTCIR12-MathWiki-10rate: 1

n = 0 1 ( n + a ) superscript subscript n 0 1 n a \sum_{n=0}^{\infty}\frac{1}{(n+a)}

NTCIR12-MathWiki-10rate: 0

F q p ( a 1 , , a p ; b 1 , , b q ; z ) = n = 0 ( a 1 ) n ( a p ) n ( b 1 ) n ( b q ) n z n n ! subscript subscript F q p subscript a 1 normal-… subscript a p subscript b 1 normal-… subscript b q z superscript subscript n 0 subscript subscript a 1 n normal-… subscript subscript a p n subscript subscript b 1 n normal-… subscript subscript b q n superscript z n n \,{}_{p}F_{q}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};z)=\sum_{n=0}^{\infty}% \frac{(a_{1})_{n}\dots(a_{p})_{n}}{(b_{1})_{n}\dots(b_{q})_{n}}\,\frac{z^{n}}{% n!}

NTCIR12-MathWiki-10rate: 1

L ( χ , s ) = n = 1 χ ( n ) n s L χ s superscript subscript n 1 χ n superscript n s L(\chi,s)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}

NTCIR12-MathWiki-10rate: 0

F q p ( a 1 , , a p ; c 1 , , c q ; z ) = n = 0 ( a 1 ) n ( a p ) n ( c 1 ) n ( c q ) n z n n ! subscript subscript F q p subscript a 1 normal-… subscript a p subscript c 1 normal-… subscript c q z superscript subscript n 0 subscript subscript a 1 n normal-⋯ subscript subscript a p n subscript subscript c 1 n normal-⋯ subscript subscript c q n superscript z n n {}_{p}F_{q}(a_{1},\dots,a_{p};c_{1},\dots,c_{q};z)=\sum_{n=0}^{\infty}\frac{(a% _{1})_{n}\cdots(a_{p})_{n}}{(c_{1})_{n}\cdots(c_{q})_{n}}\frac{z^{n}}{n!}

NTCIR12-MathWiki-10rate: 1

x = n = 0 a n 10 n . x superscript subscript n 0 subscript a n superscript 10 n x=\sum_{n=0}^{\infty}\frac{a_{n}}{10^{n}}.

NTCIR12-MathWiki-10rate: 1

ζ ( s , q ) = n = 0 1 ( q + n ) s . ζ s q superscript subscript n 0 1 superscript q n s \zeta(s,q)=\sum_{n=0}^{\infty}\frac{1}{(q+n)^{s}}.

NTCIR12-MathWiki-10rate: 2

Φ ( z , s , q ) = k = 0 z k ( k + q ) s normal-Φ z s q superscript subscript k 0 superscript z k superscript k q s \Phi(z,s,q)=\sum_{k=0}^{\infty}\frac{z^{k}}{(k+q)^{s}}

NTCIR12-MathWiki-10rate: 1

cosh x = n = 0 x 2 n ( 2 n ) ! x superscript subscript n 0 superscript x 2 n 2 n \textstyle\cosh x=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}

NTCIR12-MathWiki-10rate: 1

F 1 2 ( a , b ; c ; z ) = n = 0 ( a ) n ( b ) n ( c ) n z n n ! . subscript subscript F 1 2 a b c z superscript subscript n 0 subscript a n subscript b n subscript c n superscript z n n {}_{2}F_{1}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^% {n}}{n!}.

NTCIR12-MathWiki-10rate: 4

L ( λ , α , s ) = n = 0 exp ( 2 π i λ n ) ( n + α ) s . L λ α s superscript subscript n 0 2 π i λ n superscript n α s L(\lambda,\alpha,s)=\sum_{n=0}^{\infty}\frac{\exp(2\pi i\lambda n)}{(n+\alpha)% ^{s}}.

NTCIR12-MathWiki-10rate: 1

ζ ( 2 s ) ζ ( s ) = n = 1 λ ( n ) n s . ζ 2 s ζ s superscript subscript n 1 λ n superscript n s \frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}.

NTCIR12-MathWiki-10rate: 1

f ( x ) = k = 1 sin ( 2 k x ) 2 k f x superscript subscript k 1 superscript 2 k x superscript 2 k \displaystyle f(x)=\sum_{k=1}^{\infty}\frac{\sin(2^{k}x)}{\sqrt{2}^{k}}

NTCIR12-MathWiki-10rate: 1

Z ( P , Q , s ) = n = 1 f n ( P ) f n ( Q ) λ n s Z P Q s superscript subscript n 1 subscript f n P subscript f n Q superscript subscript λ n s Z(P,Q,s)=\sum_{n=1}^{\infty}\frac{f_{n}(P)f_{n}(Q)}{\lambda_{n}^{s}}

NTCIR12-MathWiki-10rate: 1

L ( E , s ) = n = 1 a n n s . L E s superscript subscript n 1 subscript a n superscript n s L(E,s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.

NTCIR12-MathWiki-10rate: 1

L ( s , E ) = n = 1 a n n s . L s E superscript subscript n 1 subscript a n superscript n s L(s,E)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.

NTCIR12-MathWiki-10rate: 2

ζ ( s , t ) = n = 1 H n , t ( n + 1 ) s ζ s t superscript subscript n 1 subscript H n t superscript n 1 s \zeta(s,t)=\sum_{n=1}^{\infty}\frac{H_{n,t}}{(n+1)^{s}}

NTCIR12-MathWiki-10rate: 0

exp ( z ) = n = 0 z n n ! . z superscript subscript n 0 superscript z n n \exp(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}.

NTCIR12-MathWiki-10rate: 1

n = 0 P ( n ) α n = α 2 ( α + 1 ) α 3 - α - 1 . superscript subscript n 0 P n superscript α n superscript α 2 α 1 superscript α 3 α 1 \sum_{n=0}^{\infty}\frac{P(n)}{\alpha^{n}}=\frac{\alpha^{2}(\alpha+1)}{\alpha^% {3}-\alpha-1}.

NTCIR12-MathWiki-10rate: 1

g ( s ) = n = 1 a ( n ) n s g s superscript subscript n 1 a n superscript n s g(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}

NTCIR12-MathWiki-10rate: 1

k = 0 ( - 1 ) k ( z + k ) m + 1 = 1 ( - 2 ) m + 1 m ! [ ψ ( m ) ( z 2 ) - ψ ( m ) ( z + 1 2 ) ] superscript subscript k 0 superscript 1 k superscript z k m 1 1 superscript 2 m 1 m delimited-[] superscript ψ m z 2 superscript ψ m z 1 2 \sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)^{m+1}}=\frac{1}{(-2)^{m+1}m!}\left[% \psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right]

NTCIR12-MathWiki-10rate: 1

ζ ( s ) = n = 1 1 n s , ζ s superscript subscript n 1 1 superscript n s \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},

NTCIR12-MathWiki-10rate: 1

e q x = n = 0 x n [ n ] q ! . superscript subscript e q x superscript subscript n 0 superscript x n subscript delimited-[] n q e_{q}^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{[n]_{q}!}.

NTCIR12-MathWiki-10rate: 1

A ( w ) e q ( z w ) = n = 0 p n ( z ) [ n ] q ! w n A w subscript e q z w superscript subscript n 0 subscript p n z subscript delimited-[] n q superscript w n A(w)e_{q}(zw)=\sum_{n=0}^{\infty}\frac{p_{n}(z)}{[n]_{q}!}w^{n}

NTCIR12-MathWiki-10rate: 1

ζ ( s , a ) = n = 0 1 ( n + a ) s ζ s a superscript subscript n 0 1 superscript n a s \zeta(s,a)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}\!

NTCIR12-MathWiki-10rate: 2

Φ ( z , s , q ) = k = 0 z k ( k + q ) s normal-Φ z s q superscript subscript k 0 superscript z k superscript k q s \Phi(z,s,q)=\sum_{k=0}^{\infty}\frac{z^{k}}{(k+q)^{s}}\!

NTCIR12-MathWiki-10rate: 1

L ( s , Δ ) = n = 1 a n n s L s normal-Δ superscript subscript n 1 subscript a n superscript n s L(s,\Delta)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}

NTCIR12-MathWiki-10rate: 1

ζ ( s ) = n = 1 1 n s . ζ s superscript subscript n 1 1 superscript n s \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.

NTCIR12-MathWiki-10rate: 1

cos ( φ ) = n = 0 ( - φ 2 ) n ( 2 n ) ! , φ superscript subscript n 0 superscript superscript φ 2 n 2 n \cos(\varphi)=\sum_{n=0}^{\infty}\frac{(-\varphi^{2})^{n}}{(2n)!},

NTCIR12-MathWiki-10rate: 1

G ( x , z ; λ ) = i = 1 n e i ( x ) f i * ( z ) λ - λ i . G x z λ superscript subscript i 1 n subscript e i x superscript subscript f i z λ subscript λ i G(x,z;\lambda)=\sum_{i=1}^{n}\frac{e_{i}(x)f_{i}^{*}(z)}{\lambda-\lambda_{i}}.

NTCIR12-MathWiki-10rate: 1

θ ^ F ( z ) = k = 0 R F ( k ) exp ( 2 π i k z ) , subscript normal-^ θ F z superscript subscript k 0 subscript R F k 2 π i k z \widehat{\theta}_{F}(z)=\sum_{k=0}^{\infty}R_{F}(k)\exp(2\pi ikz),

NTCIR12-MathWiki-10rate: 1

ψ 1 ( z ) = n = 0 1 ( z + n ) 2 , subscript ψ 1 z superscript subscript n 0 1 superscript z n 2 \psi_{1}(z)=\sum_{n=0}^{\infty}\frac{1}{(z+n)^{2}},

NTCIR12-MathWiki-10rate: 1

F ( s ) = n = 1 f ( n ) n s F s superscript subscript n 1 f n superscript n s F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 , a 0 formulae-sequence a superscript x 2 b x c 0 a 0 \textstyle{ax^{2}+bx+c=0,a\neq 0}

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 , a superscript x 2 b x c 0 ax^{2}+bx+c=0,

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0\,

NTCIR12-MathWiki-11rate: 3

a x 2 + b x = c a superscript x 2 b x c ax^{2}+bx=c

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 , a superscript x 2 b x c 0 ax^{2}+bx+c=0,\,

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 , with a 0 , formulae-sequence a superscript x 2 b x c 0 with a 0 ax^{2}+bx+c=0,\,\text{ with }a\neq 0,

NTCIR12-MathWiki-11rate: 4

a x 3 + b x + c = 0 a superscript x 3 b x c 0 ax^{3}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a s 2 + b s + c = 0 a superscript s 2 b s c 0 as^{2}+bs+c=0\,

NTCIR12-MathWiki-11rate: 4

a d 2 + b d + c = 0 a superscript d 2 b d c 0 ad^{2}+bd+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 \ ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 . a superscript x 2 b x c 0 . ax^{2}+bx+c=0\ .

NTCIR12-MathWiki-11rate: 2

a x + b y + c = 0 a x b y c 0 ax+by+c=0\,

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0\,

NTCIR12-MathWiki-11rate: 4

A x 2 + B x + C = 0 A superscript x 2 B x C 0 \ Ax^{2}+Bx+C=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 , a superscript x 2 b x c 0 ax^{2}+bx+c=0,\,\!

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0. a superscript x 2 b x c 0. ax^{2}+bx+c=0.

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a n 2 + b n + q = 0 a superscript n 2 b n q 0 an^{2}+bn+q=0

NTCIR12-MathWiki-11rate: 2

a x + b y + c = 0 a x b y c 0 ax+by+c=0\,

NTCIR12-MathWiki-11rate: 4

a λ 2 + b λ + c = 0 a superscript λ 2 b λ c 0 a\lambda^{2}+b\lambda+c=0\;

NTCIR12-MathWiki-11rate: 2

a x + b y + c = 0 a x b y c 0 ax+by+c=0\,

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 , a superscript x 2 b x c 0 ax^{2}+bx+c=0,\,\!

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-11rate: 4

a x 2 + b x + c = 0 a superscript x 2 b x c 0 ax^{2}+bx+c=0

NTCIR12-MathWiki-12rate: 2

O ( log n ) O n O(\log n)

NTCIR12-MathWiki-12rate: 1

O ( m n ) O m n O(mn)

NTCIR12-MathWiki-12rate: 1

O ( m n ) O m n O(mn)

NTCIR12-MathWiki-12rate: 0

Θ ( m n p ) normal-Θ m n p \Theta(mnp)

NTCIR12-MathWiki-12rate: 3

O ( η * log n ) O η n O(\eta*\log{n})

NTCIR12-MathWiki-12rate: 1

O ( m n ) O m n O(mn)

NTCIR12-MathWiki-12rate: 4

O ( V E log V ) O V E V O(VE\log V)

NTCIR12-MathWiki-12rate: 2

O ( log n ) O n O(\log n)

NTCIR12-MathWiki-12rate: 2

O ( log n ) O n O(\log n)

NTCIR12-MathWiki-12rate: 0

O ( l g n ) O l g n O(lgn)

NTCIR12-MathWiki-12rate: 0

O ( l g n ) O l g n O(lgn)

NTCIR12-MathWiki-12rate: 3

O ( n log k ) O n k O(n\log k)

NTCIR12-MathWiki-12rate: 3

O ( n k log k ) O n k k O(nk\log k)

NTCIR12-MathWiki-12rate: 2

Q D ( n ) = O ( Q S ( n ) log ( n ) ) subscript Q D n O subscript Q S n n Q_{D}\left(n\right)=O(Q_{S}\left(n\right)\log\left(n\right))\,\!

NTCIR12-MathWiki-12rate: 3

O ( E log V ) O E V O(E\log V)

NTCIR12-MathWiki-12rate: 1

O ( m r ) O m r O(mr)\,

NTCIR12-MathWiki-12rate: 4

s f 2 ( n ) = O ( M ( n ) log n ) . subscript s subscript f 2 n O M n n s_{f_{2}}(n)=O\left(M(n)\log n\right).

NTCIR12-MathWiki-12rate: 3

O ( M ( m ) log 2 m ) = O ( M ( n ) log n ) . O M m superscript 2 m O M n n O\left(M(m)\log^{2}m\right)=O\left(M(n)\log n\right).\,

NTCIR12-MathWiki-12rate: 1

O ( M S T ) O M S T O(MST)

NTCIR12-MathWiki-12rate: 3

O ( n log h ) O n h O(n\log h)

NTCIR12-MathWiki-12rate: 1

O ( K N M ) O K N M O(KNM)

NTCIR12-MathWiki-12rate: 4

O ( K N log N ) O K N N O(KN\log N)

NTCIR12-MathWiki-12rate: 1

O ( m g ( k ) n k ) O superscript m g k superscript n k O\left(m^{g(k)}n^{k}\right)

NTCIR12-MathWiki-12rate: 3

O ( d log n ) O d n O(d\log{n})

NTCIR12-MathWiki-12rate: 3

O ( n k log ( n ) ) O n k n O(nk\,\log(n))

NTCIR12-MathWiki-12rate: 4

O ( m n log m ) O m n m O(mn\log m)

NTCIR12-MathWiki-12rate: 0

O ~ ( m ) normal-~ O m \tilde{O}(m)

NTCIR12-MathWiki-12rate: 3

O ( m n log p ) = O ( n log n ) O m n p O n n O(mn\log p)=O(n\log n)

NTCIR12-MathWiki-12rate: 3

O ( k log n ) O k n O(k\log n)

NTCIR12-MathWiki-12rate: 2

O ( T m ) = O ( n 2 m log n ) O T m O superscript n 2 m n O(Tm)=O(n^{2}m\log n)

NTCIR12-MathWiki-12rate: 3

O ( m log n log log n ) O m n n O(m\log n\log\log n)

NTCIR12-MathWiki-12rate: 3

O ( λ δ ( n ) log n ) O subscript λ δ n n O(\lambda_{\delta}(n)\log n)

NTCIR12-MathWiki-12rate: 3

O ( n log m ) O n m O(n\log m)

NTCIR12-MathWiki-12rate: 3

O ( n log m ) O n m O(n\log m)

NTCIR12-MathWiki-12rate: 3

O ( m log n ) O m n O(m\log n)

NTCIR12-MathWiki-12rate: 1

O ( m + log n ) O m n O(m+\log n)

NTCIR12-MathWiki-12rate: 1

O ( d 5 n log 3 B ) O superscript d 5 n superscript 3 B O(d^{5}n\log^{3}B)\,

NTCIR12-MathWiki-12rate: 0

O ( m n p ) O m n p O(mnp)

NTCIR12-MathWiki-12rate: 2

O ( log 2 n log log n ) O superscript 2 n n O(\log^{2}n\log\log n)

NTCIR12-MathWiki-12rate: 1

O ( V E log V log ( V C ) ) O V E V V C O(VE\log V\log(VC))

NTCIR12-MathWiki-12rate: 3

O ( n log n ) O n n O(n\log n)

NTCIR12-MathWiki-12rate: 1

O ( m n ) O m n O(mn)

NTCIR12-MathWiki-12rate: 0

O ( N s κ ) O N s κ O(Ns\kappa)

NTCIR12-MathWiki-12rate: 3

O ( m n ) = O ( n 3 log n ) O m n O superscript n 3 n O(mn)=O(n^{3}\log n)

NTCIR12-MathWiki-12rate: 0

O ( c k N ) O c k N O(ckN)

NTCIR12-MathWiki-12rate: 3

O ( n log k ) O n k O(n\log k)

NTCIR12-MathWiki-12rate: 2

O ( m n r 2 log 1 ϵ ) O m n superscript r 2 1 ϵ O\left(mnr^{2}\log\frac{1}{\epsilon}\right)

NTCIR12-MathWiki-12rate: 2

O ( p n ( m + n log n ) ) O p n m n n O(pn(m+n\log n))

NTCIR12-MathWiki-12rate: 2

O ( log n ) O n O(\log n)

NTCIR12-MathWiki-12rate: 2

O ( log n ) O n O(\log n)

NTCIR12-MathWiki-12rate: 2

O ( log n ) O n O(\log{n})

NTCIR12-MathWiki-12rate: 2

O ( log n ) O n O(\log{n})

NTCIR12-MathWiki-12rate: 2

T amortized ( m ) = O ( m log n ) subscript T amortized m O m n T_{\mathrm{amortized}}(m)=O(m\log n)

NTCIR12-MathWiki-12rate: 0

O ( m ) O m O(m)

NTCIR12-MathWiki-12rate: 0

O ( m ) O m O(m)

NTCIR12-MathWiki-12rate: 0

O ( m ) O m O(m)

NTCIR12-MathWiki-12rate: 0

O ( m ) O m O(m)

NTCIR12-MathWiki-12rate: 1

O ( m n ) O m n O(mn)

NTCIR12-MathWiki-12rate: 2

O ( m n log n ) O m n n O(m\sqrt{n}\log n)

NTCIR12-MathWiki-13rate: 0

I n f = ( A n f ) 2 , superscript subscript I n f superscript superscript subscript A n f 2 I_{n}^{f}=\left(A_{n}^{f}\right)^{2},

NTCIR12-MathWiki-13rate: 0

A = ( r n - 1 ) / B = 3 A superscript r n 1 B 3 A=(r^{n}-1)/B=3

NTCIR12-MathWiki-13rate: 0

R = ( A + I ) n R superscript A I n R=(A+I)^{n}

NTCIR12-MathWiki-13rate: 0

M 1 = M P = ( M P ) ( P M ) subscript M 1 direct-sum M P M P P M M_{1}=M\oplus P=(M\setminus P)\cup(P\setminus M)

NTCIR12-MathWiki-13rate: 1

x y = ( x y ) ¬ ( x y ) direct-sum x y x y x y x\oplus y=(x\vee y)\wedge\neg{(x\wedge y)}

NTCIR12-MathWiki-13rate: 0

P = P Δ superscript P normal-′ direct-sum P normal-Δ P^{\prime}=P\oplus\Delta

NTCIR12-MathWiki-13rate: 0

P ( A , B ) = A B superscript P normal-′ A B direct-sum A B P^{\prime}(A,B)=A\oplus B

NTCIR12-MathWiki-13rate: 0

( A × B ) c = ( A c × B c ) ( A c × B ) ( A × B c ) . superscript A B c superscript A c superscript B c superscript A c B A superscript B c (A\times B)^{c}=(A^{c}\times B^{c})\cup(A^{c}\times B)\cup(A\times B^{c}).

NTCIR12-MathWiki-13rate: 0

A B = ( A B ) B , normal-∙ A B symmetric-difference direct-sum A B B A\bullet B=(A\oplus B)\ominus B,\,

NTCIR12-MathWiki-13rate: 0

B = ( x b i ) B superscript x subscript b i \mathit{B}=(x^{b_{i}})

NTCIR12-MathWiki-13rate: 0

x y = ( x + y ) / 2 direct-sum x y x y 2 x\oplus y=(x+y)/2

NTCIR12-MathWiki-13rate: 0

( A B ) c = A c B c . superscript A B c superscript A c superscript B c \left(A\cup B\right)^{c}=A^{c}\cap B^{c}.

NTCIR12-MathWiki-13rate: 0

𝔤 = 𝔨 𝔭 𝔤 direct-sum 𝔨 𝔭 \displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}}

NTCIR12-MathWiki-13rate: 0

Y = Y N superscript Y normal-′ direct-sum Y N Y^{\prime}=Y\oplus N

NTCIR12-MathWiki-13rate: 0

A B = A ¯ B ¯ ¯ A B normal-¯ normal-¯ A normal-¯ B A\cap B=\overline{\overline{A}\cup\overline{B}}

NTCIR12-MathWiki-13rate: 0

M = T P M direct-sum T P M=T\oplus P

NTCIR12-MathWiki-13rate: 1

A c B c ( A B ) c superscript A c superscript B c superscript A B c A^{c}\cup B^{c}\subseteq(A\cap B)^{c}

NTCIR12-MathWiki-13rate: 1

A c B c ( A B ) c superscript A c superscript B c superscript A B c A^{c}\cup B^{c}\subseteq(A\cap B)^{c}

NTCIR12-MathWiki-13rate: 1

A c B c ( A B ) c superscript A c superscript B c superscript A B c A^{c}\cup B^{c}\subseteq(A\cap B)^{c}

NTCIR12-MathWiki-13rate: 0

( A B ) c = A c B c superscript A B c superscript A c superscript B c (A\cup B)^{c}=A^{c}\cap B^{c}

NTCIR12-MathWiki-13rate: 1

A B = { z E ( B s ) z A } direct-sum A B conditional-set z E subscript superscript B s z A A\oplus B=\{z\in E\mid(B^{s})_{z}\cap A\neq\varnothing\}

NTCIR12-MathWiki-13rate: 0

Q = ( A T A ) - 1 Q superscript superscript A T A 1 Q=\left(A^{T}A\right)^{-1}

NTCIR12-MathWiki-13rate: 0

B c A c superscript B c superscript A c B^{c}\subseteq A^{c}

NTCIR12-MathWiki-13rate: 0

A B = ( A B + B A ) / 2 A B A B B A 2 A\circ B=(AB+BA)/2

NTCIR12-MathWiki-13rate: 0

K = [ R ] ρ [ S ] σ [ A ] α [ B ] β × γ R ρ γ S σ γ A α γ B β × ( c A ) α ( c B ) β ( c R ) ρ ( c S ) σ = Q E Γ C 0 superscript K symmetric-difference superscript delimited-[] R ρ superscript delimited-[] S σ normal-… superscript delimited-[] A α superscript delimited-[] B β normal-… superscript subscript γ R ρ superscript subscript γ S σ normal-… superscript subscript γ A α superscript subscript γ B β normal-… superscript subscript superscript c symmetric-difference A α superscript subscript superscript c symmetric-difference B β normal-… superscript subscript superscript c symmetric-difference R ρ superscript subscript superscript c symmetric-difference S σ normal-… superscript Q E normal-Γ superscript C 0 K^{\ominus}=\frac{[R]^{\rho}[S]^{\sigma}...}{[A]^{\alpha}[B]^{\beta}...}\times% \frac{{\gamma_{R}}^{\rho}{\gamma_{S}}^{\sigma}...}{{\gamma_{A}}^{\alpha}{% \gamma_{B}}^{\beta}...}\times\frac{\left({c^{\ominus}_{A}}\right)^{\alpha}% \left({c^{\ominus}_{B}}\right)^{\beta}...}{\left({c^{\ominus}_{R}}\right)^{% \rho}\left({c^{\ominus}_{S}}\right)^{\sigma}...}=Q^{E}\Gamma C^{0}

NTCIR12-MathWiki-13rate: 0

( A B ) C = A ( B C ) symmetric-difference symmetric-difference A B C symmetric-difference A direct-sum B C (A\ominus B)\ominus C=A\ominus(B\oplus C)

NTCIR12-MathWiki-13rate: 0

Q = ( A T A ) - 1 Q superscript superscript A T A 1 Q=\left(A^{T}A\right)^{-1}

NTCIR12-MathWiki-13rate: 0

( A B ) direct-sum A B (A\oplus B)

NTCIR12-MathWiki-13rate: 0

A B := ( A B ) B . fragments A B assign fragments normal-( A normal-→ B normal-) normal-→ B normal-. \ A\vee B:=(A\rightarrow B)\rightarrow B.

NTCIR12-MathWiki-13rate: 0

I p , = J K superscript I p normal-∙ direct-sum J K I^{p,\bullet}=J\oplus K

NTCIR12-MathWiki-13rate: 1

gyr [ 𝐮 , 𝐯 ] 𝐰 = ( 𝐮 𝐯 ) ( 𝐮 ( 𝐯 𝐰 ) ) gyr 𝐮 𝐯 𝐰 direct-sum symmetric-difference direct-sum 𝐮 𝐯 direct-sum 𝐮 direct-sum 𝐯 𝐰 \mathrm{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{% v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))

NTCIR12-MathWiki-13rate: 0

A B = ( A C ) ( A c D ) direct-product A B symmetric-difference A C symmetric-difference superscript A c D A\odot B=(A\ominus C)\cap(A^{c}\ominus D)

NTCIR12-MathWiki-13rate: 0

x = ( A T A ) - 1 A T b . x superscript superscript A T A 1 superscript A T b x=(A^{T}A)^{-1}A^{T}b.

NTCIR12-MathWiki-13rate: 1

A B = ( A B ) - ( A B ) A normal-△ B A B A B A\triangle B=(A\cup B)-(A\cap B)

NTCIR12-MathWiki-13rate: 0

U = ( A T A ) 1 / 2 A - 1 U superscript superscript A T A 1 2 superscript A 1 U=(A^{T}A)^{1/2}A^{-1}

NTCIR12-MathWiki-13rate: 0

α β := α ( β ) assign symmetric-difference α β direct-sum α symmetric-difference β \alpha\ominus\beta:=\alpha\oplus(\ominus\beta)

NTCIR12-MathWiki-13rate: 0

ϵ = ( log log n ) c ϵ superscript n c \epsilon=(\log\log n)^{c}

NTCIR12-MathWiki-13rate: 0

A B = ( A B ) & ( B A ) fragments A B fragments normal-( A normal-⊸ B normal-) fragments normal-( B normal-⊸ A normal-) A\equiv B\quad=\quad(A\multimap B)\&(B\multimap A)

NTCIR12-MathWiki-13rate: 0

w = ( A T A ) - 1 A T b . w superscript superscript A T A 1 superscript A T b w=(A^{T}A)^{-1}A^{T}b.

NTCIR12-MathWiki-13rate: 0

B ( A B ) normal-→ B A B B\to(A\lor B)

NTCIR12-MathWiki-13rate: 0

B ( A B ) normal-→ B A B B\to(A\lor B)

NTCIR12-MathWiki-13rate: 0

A B ¯ normal-¯ direct-sum A B \overline{A\oplus B}

NTCIR12-MathWiki-13rate: 1

gyr [ 𝐮 , 𝐯 ] 𝐰 = ( 𝐮 𝐯 ) ( 𝐮 ( 𝐯 𝐰 ) ) gyr 𝐮 𝐯 𝐰 direct-sum symmetric-difference direct-sum 𝐮 𝐯 direct-sum 𝐮 direct-sum 𝐯 𝐰 \,\text{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{% v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))

NTCIR12-MathWiki-13rate: 1

v = ( A T A ) - 1 A T b normal-v superscript superscript A T A 1 superscript A T b \mathrm{v}=(A^{T}A)^{-1}A^{T}b

NTCIR12-MathWiki-13rate: 2

A B = ( A c B s ) c direct-sum A B superscript symmetric-difference superscript A c superscript B s c A\oplus B=(A^{c}\ominus B^{s})^{c}

NTCIR12-MathWiki-13rate: 1

W = ( A T A ) - 1 A T W superscript superscript A T A 1 superscript A T W=(A^{T}A)^{-1}A^{T}

NTCIR12-MathWiki-13rate: 2

A B = ( A c B s ) c normal-∙ A B superscript superscript A c superscript B s c A\bullet B=(A^{c}\circ B^{s})^{c}

NTCIR12-MathWiki-13rate: 0

V = U W V direct-sum U W V=U\oplus W

NTCIR12-MathWiki-13rate: 0

A = R V , A direct-sum R V A=R\oplus V,

NTCIR12-MathWiki-13rate: 0

A B = ( A B ) ( B A ) A normal-△ B A B B A A\,\triangle\,B=(A\setminus B)\cup(B\setminus A)

NTCIR12-MathWiki-13rate: 0

A B = ( A B ) ( A B ) A B A normal-△ B normal-△ A B A\cup B=(A\,\triangle\,B)\,\triangle\,(A\cap B)

NTCIR12-MathWiki-13rate: 0

𝔤 = 𝔨 𝔭 𝔤 direct-sum 𝔨 𝔭 \mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}

NTCIR12-MathWiki-13rate: 0

A B = 𝒩 . direct-sum A B 𝒩 A\oplus B=\mathcal{N}.

NTCIR12-MathWiki-13rate: 0

A Δ B = ( A B ) ( B A ) . A normal-Δ B A B B A A\,\Delta\,B=(A\setminus B)\cup(B\setminus A).

NTCIR12-MathWiki-13rate: 2

π σ = rev ( rev ( σ ) rev ( π ) ) direct-sum π σ rev symmetric-difference rev σ rev π \pi\oplus\sigma=\operatorname{rev}(\operatorname{rev}(\sigma)\ominus% \operatorname{rev}(\pi))

NTCIR12-MathWiki-13rate: 0

D = y x D direct-sum y x D=y\oplus x

NTCIR12-MathWiki-13rate: 0

A B = ( A B ) ( A B ) A B A normal-△ B normal-△ A B A\,\cup\,B=(A\,\triangle\,B)\,\triangle\,(A\cap B)

NTCIR12-MathWiki-13rate: 0

A B . direct-sum A B A\oplus B.

NTCIR12-MathWiki-13rate: 0

R G = ( dom G R ) G fragments R direct-sum G fragments normal-( dom G not-subgroup-of R normal-) G R\oplus G=(\,\text{dom }G\ntriangleleft R)\cup G

NTCIR12-MathWiki-13rate: 0

C = A B C direct-sum A B C=A\oplus B

NTCIR12-MathWiki-13rate: 0

A B = A ¯ + B ¯ ¯ normal-⋅ A B normal-¯ normal-¯ A normal-¯ B A\cdot B=\overline{\overline{A}+\overline{B}}

NTCIR12-MathWiki-13rate: 1

A B = ( A C B C ) C A B superscript superscript A C superscript B C C A\cup B=\left(A^{C}\cap B^{C}\right)^{C}

NTCIR12-MathWiki-13rate: 0

( A B ) A = A ( A B ) direct-sum direct-sum A B A direct-sum A direct-sum A B (A\oplus B)\oplus A=A\oplus(A\oplus B)

NTCIR12-MathWiki-13rate: 0

A B = B A direct-sum A B direct-sum B A A\oplus B=B\oplus A

NTCIR12-MathWiki-14rate: 1

s y m b o l ω = ( α ˙ sin β sin γ + β ˙ cos γ ) I + ( α ˙ sin β cos γ - β ˙ sin γ ) J + ( α ˙ cos β + γ ˙ ) K s y m b o l ω normal-˙ α β γ normal-˙ β γ I normal-˙ α β γ normal-˙ β γ J normal-˙ α β normal-˙ γ K symbol\omega=(\dot{\alpha}\sin\beta\sin\gamma+\dot{\beta}\cos\gamma)I+(\dot{% \alpha}\sin\beta\cos\gamma-\dot{\beta}\sin\gamma)J+(\dot{\alpha}\cos\beta+\dot% {\gamma})K

NTCIR12-MathWiki-14rate: 0

F t o t a l = μ m g cos θ + m g sin θ subscript F t o t a l μ m g θ m g θ F_{total}=\mu mg\cos{\theta}+mg\sin{\theta}

NTCIR12-MathWiki-14rate: 0

cos C = - cos A C A \cos C=-\cos A

NTCIR12-MathWiki-14rate: 0

a b c 1 - cos 2 α - cos 2 β - cos 2 γ + 2 cos α cos β cos γ a b c 1 superscript 2 α superscript 2 β superscript 2 γ 2 α β γ abc\sqrt{1-\cos^{2}\alpha-\cos^{2}\beta-\cos^{2}\gamma+2\cos\alpha\cos\beta% \cos\gamma}

NTCIR12-MathWiki-14rate: 1

y ( u , v ) = - cos θ sinh v cos u + sin θ cosh v sin u y u v θ v u θ v u y(u,v)=-\cos\theta\,\sinh v\,\cos u+\sin\theta\,\cosh v\,\sin u

NTCIR12-MathWiki-14rate: 0

tan λ = sin α cos ε + tan δ sin ε cos α ; { cos β sin λ = cos δ sin α cos ε + sin δ sin ε ; cos β cos λ = cos δ cos α . λ α ε δ ε α cases β λ δ α ε δ ε otherwise β λ δ α otherwise \tan\lambda={\sin\alpha\cos\varepsilon+\tan\delta\sin\varepsilon\over\cos% \alpha};\qquad\qquad\begin{cases}\cos\beta\sin\lambda=\cos\delta\sin\alpha\cos% \varepsilon+\sin\delta\sin\varepsilon;\\ \cos\beta\cos\lambda=\cos\delta\cos\alpha.\end{cases}

NTCIR12-MathWiki-14rate: 1

sin δ = sin β cos ε + cos β sin ε sin λ δ β ε β ε λ \sin\delta=\sin\beta\cos\varepsilon+\cos\beta\sin\varepsilon\sin\lambda

NTCIR12-MathWiki-14rate: 1

sin a = sin ϕ o sin δ + cos ϕ o cos δ cos h a subscript ϕ o δ subscript ϕ o δ h \sin a=\sin\phi_{o}\sin\delta+\cos\phi_{o}\cos\delta\cos h

NTCIR12-MathWiki-14rate: 0

d G T , P = - k 𝔸 k d ξ k + W . d subscript G T P subscript k subscript 𝔸 k d subscript ξ k superscript W normal-′ dG_{T,P}=-\sum_{k}\mathbb{A}_{k}\,d\xi_{k}+W^{\prime}.\,

NTCIR12-MathWiki-14rate: 0

sin ( x + y ) = sin x cos y + cos x sin y , x y x y x y \,\sin(x+y)=\sin x\cos y+\cos x\sin y,\,

NTCIR12-MathWiki-14rate: 0

cos ( x - y ) = cos x cos y - sin x sin y x y x y x y \cos(x-y)=\cos x\cos y-\sin x\sin y

NTCIR12-MathWiki-14rate: 0

[ cos θ cos ψ - cos ϕ sin ψ + sin ϕ sin θ cos ψ sin ϕ sin ψ + cos ϕ sin θ cos ψ cos θ sin ψ cos ϕ cos ψ + sin ϕ sin θ sin ψ - sin ϕ cos ψ + cos ϕ sin θ sin ψ - sin θ sin ϕ cos θ cos ϕ cos θ ] θ ψ ϕ ψ ϕ θ ψ ϕ ψ ϕ θ ψ θ ψ ϕ ψ ϕ θ ψ ϕ ψ ϕ θ ψ θ ϕ θ ϕ θ \begin{bmatrix}\cos\theta\cos\psi&-\cos\phi\sin\psi+\sin\phi\sin\theta\cos\psi% &\sin\phi\sin\psi+\cos\phi\sin\theta\cos\psi\\ \cos\theta\sin\psi&\cos\phi\cos\psi+\sin\phi\sin\theta\sin\psi&-\sin\phi\cos% \psi+\cos\phi\sin\theta\sin\psi\\ -\sin\theta&\sin\phi\cos\theta&\cos\phi\cos\theta\\ \end{bmatrix}

NTCIR12-MathWiki-14rate: 2

cos c = cos a cos b + sin a sin b cos γ . c a b a b γ \cos c=\cos a\,\cos b+\sin a\,\sin b\,\cos\gamma.

NTCIR12-MathWiki-14rate: 0

[ 𝐚 ^ 𝐛 ^ 𝐜 ^ ] = [ 𝟏 𝐚 - cos ( γ ) 𝐚 sin ( γ ) cos ( α ) cos ( γ ) - cos ( β ) 𝐚𝐯 sin ( γ ) 𝟎 𝟏 𝐛 sin ( γ ) cos ( β ) cos ( γ ) - cos ( α ) 𝐛𝐯 sin ( γ ) 𝟎 𝟎 sin ( γ ) 𝐜𝐯 ] [ x y z ] normal-^ 𝐚 normal-^ 𝐛 normal-^ 𝐜 1 𝐚 γ 𝐚 γ α γ β 𝐚𝐯 γ 0 1 𝐛 γ β γ α 𝐛𝐯 γ 0 0 γ 𝐜𝐯 x y z \mathbf{\begin{bmatrix}\hat{a}\\ \hat{b}\\ \hat{c}\\ \end{bmatrix}=\begin{bmatrix}\frac{1}{a}&-\frac{\cos(\gamma)}{a\sin(\gamma)}&% \frac{\cos(\alpha)\cos(\gamma)-\cos(\beta)}{av\sin(\gamma)}\\ 0&\frac{1}{b\sin(\gamma)}&\frac{\cos(\beta)\cos(\gamma)-\cos(\alpha)}{bv\sin(% \gamma)}\\ 0&0&\frac{\sin(\gamma)}{cv}\\ \end{bmatrix}}\begin{bmatrix}x\\ y\\ z\\ \end{bmatrix}

NTCIR12-MathWiki-14rate: 0

σ 2 = cos ψ d θ + sin ψ sin θ d ϕ subscript σ 2 ψ d θ ψ θ d ϕ \sigma_{2}=\cos\psi\,d\theta+\sin\psi\sin\theta\,d\phi

NTCIR12-MathWiki-14rate: 1

cos γ = cos θ cos θ + sin θ sin θ cos ( φ - φ ) . γ θ superscript θ normal-′ θ superscript θ normal-′ φ superscript φ normal-′ \cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(% \varphi-\varphi^{\prime}).

NTCIR12-MathWiki-14rate: 2

cos ( c ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) cos ( C ) . c a b a b C \cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\,

NTCIR12-MathWiki-14rate: 1

sin ( a + b ) = sin a cos b + cos a sin b a b a b a b \sin\left(a+b\right)=\sin a\cos b+\cos a\sin b

NTCIR12-MathWiki-14rate: 4

cos α = - cos β cos γ + sin β sin γ cosh a k , α β γ β γ a k \cos\alpha=-\cos\beta\cos\gamma+\sin\beta\sin\gamma\cosh\frac{a}{k},\,

NTCIR12-MathWiki-14rate: 3

cos C = - cos A cos B + sin A sin B cosh c , C A B A B c \cos C=-\cos A\cos B+\sin A\sin B\cosh c,

NTCIR12-MathWiki-14rate: 0

γ α θ cos α + γ θ β cos β + γ α β = 0 subscript γ α θ α subscript γ θ β β subscript γ α β 0 \gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{% \alpha\beta}\ =0

NTCIR12-MathWiki-14rate: 3

cos A = - cos B cos C + sin B sin C cosh a . A B C B C a \cos A=-\cos B\cos C+\sin B\sin C\cosh a.\,

NTCIR12-MathWiki-14rate: 0

sin ( α + β ) = sin α cos β + cos α sin β α β α β α β \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta

NTCIR12-MathWiki-14rate: 0

cos ( α + β ) = cos α cos β - sin α sin β α β α β α β \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta

NTCIR12-MathWiki-14rate: 0

sin ( α + β ) = sin α cos β + cos α sin β α β α β α β \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta

NTCIR12-MathWiki-14rate: 0

cos ( α + β ) = cos α cos β - sin α sin β α β α β α β \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta

NTCIR12-MathWiki-14rate: 0

[ cos ( α ) cos ( γ ) - sin ( α ) sin ( β ) sin ( γ ) - sin ( α ) cos ( β ) - cos ( α ) sin ( γ ) - sin ( α ) sin ( β ) cos ( γ ) cos ( α ) sin ( β ) sin ( γ ) + sin ( α ) cos ( γ ) cos ( α ) cos ( β ) cos ( α ) sin ( β ) cos ( γ ) - sin ( α ) sin ( γ ) cos ( β ) sin ( γ ) - sin ( β ) cos ( β ) cos ( γ ) ] α γ α β γ α β α γ α β γ α β γ α γ α β α β γ α γ β γ β β γ \begin{bmatrix}\cos(\alpha)\cos(\gamma)-\sin(\alpha)\sin(\beta)\sin(\gamma)&-% \sin(\alpha)\cos(\beta)&-\cos(\alpha)\sin(\gamma)-\sin(\alpha)\sin(\beta)\cos(% \gamma)\\ \cos(\alpha)\sin(\beta)\sin(\gamma)+\sin(\alpha)\cos(\gamma)&\cos(\alpha)\cos(% \beta)&\cos(\alpha)\sin(\beta)\cos(\gamma)-\sin(\alpha)\sin(\gamma)\\ \cos(\beta)\sin(\gamma)&-\sin(\beta)&\cos(\beta)\cos(\gamma)\end{bmatrix}

NTCIR12-MathWiki-14rate: 0

cos θ 1 cos θ 2 + sin θ 1 sin θ 2 = 0 subscript θ 1 subscript θ 2 subscript θ 1 subscript θ 2 0 \cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}=0

NTCIR12-MathWiki-14rate: 0

p i = cos Ω cos ω - sin Ω cos i sin ω p j = sin Ω cos ω + cos Ω cos i sin ω p k = sin i sin ω q i = - cos Ω sin ω - sin Ω cos i cos ω q j = - sin Ω sin ω + cos Ω cos i cos ω q k = sin i cos ω w i = sin i sin Ω w j = - sin i cos Ω w k = cos i subscript p i absent normal-Ω ω normal-Ω i ω subscript p j absent normal-Ω ω normal-Ω i ω subscript p k absent i ω subscript q i absent normal-Ω ω normal-Ω i ω subscript q j absent normal-Ω ω normal-Ω i ω subscript q k absent i ω subscript w i absent i normal-Ω subscript w j absent i normal-Ω subscript w k absent i \begin{aligned}\displaystyle p_{i}&\displaystyle=\cos\Omega\cos\omega-\sin% \Omega\cos i\sin\omega\\ \displaystyle p_{j}&\displaystyle=\sin\Omega\cos\omega+\cos\Omega\cos i\sin% \omega\\ \displaystyle p_{k}&\displaystyle=\sin i\sin\omega\\ \displaystyle q_{i}&\displaystyle=-\cos\Omega\sin\omega-\sin\Omega\cos i\cos% \omega\\ \displaystyle q_{j}&\displaystyle=-\sin\Omega\sin\omega+\cos\Omega\cos i\cos% \omega\\ \displaystyle q_{k}&\displaystyle=\sin i\cos\omega\\ \displaystyle w_{i}&\displaystyle=\sin i\sin\Omega\\ \displaystyle w_{j}&\displaystyle=-\sin i\cos\Omega\\ \displaystyle w_{k}&\displaystyle=\cos i\end{aligned}

NTCIR12-MathWiki-14rate: 0

cos ( A + B ) = cos ( A ) cos ( B ) + sin ( A ) sin ( B ) A B A B A B \cos(A+B)=\cos(A)\cos(B)+\sin(A)\sin(B)\,

NTCIR12-MathWiki-14rate: 0

u φ = - u x r sin φ + u y r cos φ = - y u x + x u y . u φ u x r φ u y r φ y u x x u y \frac{\partial u}{\partial\varphi}=-\frac{\partial u}{\partial x}r\sin\varphi+% \frac{\partial u}{\partial y}r\cos\varphi=-y\frac{\partial u}{\partial x}+x% \frac{\partial u}{\partial y}.

NTCIR12-MathWiki-14rate: 0

cos ( α - β ) = cos α cos β + sin α sin β α β α β α β \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta

NTCIR12-MathWiki-14rate: 0

cos ( α + β ) + cos ( α - β ) = cos α cos β - sin α sin β + cos α cos β + sin α sin β = 2 cos α cos β α β α β α β α β α β α β 2 α β \cos(\alpha+\beta)+\cos(\alpha-\beta)=\cos\alpha\cos\beta\ -\sin\alpha\sin% \beta+\cos\alpha\cos\beta+\sin\alpha\sin\beta=2\cos\alpha\cos\beta

NTCIR12-MathWiki-14rate: 0

sin ( α + β ) = sin α cos β + sin β cos α α β α β β α \sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha

NTCIR12-MathWiki-14rate: 0

cos ( α - β ) = cos α cos β + sin α sin β α β α β α β \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\,

NTCIR12-MathWiki-14rate: 2

cos a = cos b cos c + sin b sin c cos α a b c b c α \cos a=\cos b\cos c+\sin b\sin c\cos\alpha

NTCIR12-MathWiki-14rate: 1

sin ( α + β ) sin ( β + γ ) = sin α sin γ + sin β sin ( α + β + γ ) α β β γ α γ β α β γ \sin(\alpha+\beta)\sin(\beta+\gamma)=\sin\alpha\sin\gamma+\sin\beta\sin(\alpha% +\beta+\gamma)

NTCIR12-MathWiki-14rate: 0

cos ( x + y ) = cos x cos y - sin x sin y x y x y x y \cos(x+y)=\cos{x}\cos{y}-\sin{x}\sin{y}

NTCIR12-MathWiki-14rate: 1

sin α sin β cos β cos γ + sin α cos 2 β sin γ + cos α sin 2 β cos γ + cos α sin β cos β sin γ α β β γ α superscript 2 β γ α superscript 2 β γ α β β γ \sin\alpha\sin\beta\cos\beta\cos\gamma+\sin\alpha\cos^{2}\beta\sin\gamma+\cos% \alpha\sin^{2}\beta\cos\gamma+\cos\alpha\sin\beta\cos\beta\sin\gamma

NTCIR12-MathWiki-14rate: 0

sin ( x + y ) = sin x cos y + sin y cos x x y x y y x \sin(x+y)=\sin x\,\cos y\ +\ \sin y\,\cos x

NTCIR12-MathWiki-14rate: 0

cos ( θ - α ) = cos ( θ ) cos ( α ) + sin ( θ ) sin ( α ) θ α θ α θ α \cos(\theta-\alpha)=\cos(\theta)\cos(\alpha)+\sin(\theta)\sin(\alpha)

NTCIR12-MathWiki-14rate: 0

𝐑 ( α , β , γ ) = ( cos α - sin α 0 sin α cos α 0 0 0 1 ) ( cos β 0 sin β 0 1 0 - sin β 0 cos β ) ( cos γ - sin γ 0 sin γ cos γ 0 0 0 1 ) 𝐑 α β γ α α 0 α α 0 0 0 1 β 0 β 0 1 0 β 0 β γ γ 0 γ γ 0 0 0 1 \mathbf{R}(\alpha,\beta,\gamma)=\begin{pmatrix}\cos\alpha&-\sin\alpha&0\\ \sin\alpha&\cos\alpha&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\\ \end{pmatrix}\begin{pmatrix}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{pmatrix}

NTCIR12-MathWiki-14rate: 0

sin α s = cos h cos δ cos φ + sin δ sin φ subscript α normal-s h δ φ δ φ \sin\alpha_{\mathrm{s}}=\cos h\cos\delta\cos\varphi+\sin\delta\sin\varphi

NTCIR12-MathWiki-14rate: 1

a = arccos ( cos α + cos β cos γ sin β sin γ ) , a α β γ β γ a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}% \right),

NTCIR12-MathWiki-14rate: 1

c = arccos ( cos a cos b + sin a sin b cos γ ) c a b a b γ c=\arccos\left(\cos a\cos b+\sin a\sin b\cos\gamma\right)

NTCIR12-MathWiki-14rate: 1

a = arccos ( cos α + cos β cos γ sin β sin γ ) a α β γ β γ a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}\right)

NTCIR12-MathWiki-14rate: 0

cos ( θ - θ ) = cos θ cos θ + sin θ sin θ superscript θ normal-′ θ superscript θ normal-′ θ θ superscript θ normal-′ \cos(\theta^{\prime}-\theta)=\cos\theta^{\prime}\cos\theta+\sin\theta\sin% \theta^{\prime}

NTCIR12-MathWiki-14rate: 4

cos ( A ) = - cos ( B ) cos ( C ) + sin ( B ) sin ( C ) cos ( a ) A B C B C a \cos(A)=-\cos(B)\cos(C)+\sin(B)\sin(C)\cos(a)\,

NTCIR12-MathWiki-14rate: 3

cos A = - cos B cos C + sin B sin C cos a , A B C B C a \cos A=-\cos B\,\cos C+\sin B\,\sin C\,\cos a,

NTCIR12-MathWiki-14rate: 3

cos B = - cos C cos A + sin C sin A cos b , B C A C A b \cos B=-\cos C\,\cos A+\sin C\,\sin A\,\cos b,

NTCIR12-MathWiki-14rate: 0

C cos μ + L sin μ + T ( sin α sin μ + cos α cos μ sin β ) = W g ( V cos γ ) 2 R ( x E - y E plane radial direction ) C μ L μ T α μ α μ β W g superscript V γ 2 R subscript x E - subscript y E plane radial direction C\cos{\mu}+L\sin{\mu}+T(\sin{\alpha}\sin{\mu}+\cos{\alpha}\cos{\mu}\sin{\beta}% )=\frac{W}{g}\frac{(V\cos{\gamma})^{2}}{R}\quad(x_{E}\,\text{-}y_{E}\,\text{ % plane radial direction})

NTCIR12-MathWiki-14rate: 1

sin G = cos L cos D cos R - sin L sin R = - cos L cos D sin I + sin L cos I G L D R L R L D I L I \sin G=\cos L\cos D\cos R-\sin L\sin R=-\cos L\cos D\sin I+\sin L\cos I

NTCIR12-MathWiki-14rate: 0

cos ( x - y ) = cos x cos y + sin x sin y x y x y x y \cos(x-y)=\cos x\cos y+\sin x\sin y\,

NTCIR12-MathWiki-14rate: 0

cos ( x - y ) = cos x cos y + sin x sin y . x y x y x y \cos\left(x-y\right)=\cos x\cos y+\sin x\sin y.\,

NTCIR12-MathWiki-14rate: 0

cos ( A ± B ) = cos A cos B sin A sin B plus-or-minus A B minus-or-plus A B A B \cos(A\pm B)=\cos A\ \cos B\mp\sin A\ \sin B

NTCIR12-MathWiki-14rate: 1

𝐫 ^ = sin θ cos φ 𝐱 ^ + sin θ sin φ 𝐲 ^ + cos θ 𝐳 ^ normal-^ 𝐫 θ φ normal-^ 𝐱 θ φ normal-^ 𝐲 θ normal-^ 𝐳 \mathbf{\hat{r}}=\sin\theta\cos\varphi\mathbf{\hat{x}}+\sin\theta\sin\varphi% \mathbf{\hat{y}}+\cos\theta\mathbf{\hat{z}}

NTCIR12-MathWiki-14rate: 0

𝐱 ^ 2 = [ - cos β sin γ 2 , cos γ 2 , sin β sin γ 2 ] subscript normal-^ 𝐱 2 β subscript γ 2 subscript γ 2 β subscript γ 2 \hat{\mathbf{x}}_{2}=[-\cos\beta\sin\gamma_{2}\,,\,\cos\gamma_{2}\,,\,\sin% \beta\sin\gamma_{2}]

NTCIR12-MathWiki-14rate: 0

α 2 = arctan ( sin α - sin U 1 sin σ + cos U 1 cos σ cos α 1 ) subscript α 2 α subscript U 1 σ subscript U 1 σ subscript α 1 \alpha_{2}=\arctan\left(\frac{\sin\alpha}{-\sin U_{1}\sin\sigma+\cos U_{1}\cos% \sigma\cos\alpha_{1}}\right)\,

NTCIR12-MathWiki-14rate: 0

γ α θ cos α + γ θ β cos β + γ α β = 0 subscript γ α θ α subscript γ θ β β subscript γ α β 0 \gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{% \alpha\beta}\ =0

NTCIR12-MathWiki-14rate: 0

𝒫 ^ 1 = i ( cos γ sin β α - sin γ β - cot β cos γ γ ) 𝒫 ^ 2 = i ( - sin γ sin β α - cos γ β + cot β sin γ γ ) 𝒫 ^ 3 = - i γ , subscript normal-^ 𝒫 1 i γ β α γ β β γ γ subscript normal-^ 𝒫 2 i γ β α γ β β γ γ subscript normal-^ 𝒫 3 i γ \begin{array}[]{lcl}\hat{\mathcal{P}}_{1}&=&\,i\left({\cos\gamma\over\sin\beta% }{\partial\over\partial\alpha}-\sin\gamma{\partial\over\partial\beta}-\cot% \beta\cos\gamma{\partial\over\partial\gamma}\right)\\ \hat{\mathcal{P}}_{2}&=&\,i\left(-{\sin\gamma\over\sin\beta}{\partial\over% \partial\alpha}-\cos\gamma{\partial\over\partial\beta}+\cot\beta\sin\gamma{% \partial\over\partial\gamma}\right)\\ \hat{\mathcal{P}}_{3}&=&-i{\partial\over\partial\gamma},\\ \end{array}

NTCIR12-MathWiki-15rate: 0

x , y A x y superscript A normal-∗ x,y\in A^{\ast}

NTCIR12-MathWiki-15rate: 2

x V y ( y x ) y ( y x ¬ z ( z y z x ) ) . fragments x V y fragments normal-( y x normal-) normal-→ y fragments normal-( y x z fragments normal-( z y z x normal-) normal-) normal-. x\in V\land\exists y(y\in x)\rightarrow\exists y(y\in x\land\lnot\exists z(z% \in y\land z\in x)).

NTCIR12-MathWiki-15rate: 0

x , y V , for-all x y V \forall x,y\in V,

NTCIR12-MathWiki-15rate: 0

A = { x X | ( y Y ) x , y B } . fragments A fragments normal-{ x X normal-| fragments normal-( y Y normal-) fragments normal-⟨ x normal-, y normal-⟩ B normal-} normal-. A=\{x\in X|(\exists y\in Y)\langle x,y\rangle\in B\}.

NTCIR12-MathWiki-15rate: 1

μ ( x + y ) + μ ( x ) + μ ( y ) λ ( x , y ) ( mod 2 ) x , y H k ( M ; 2 ) fragments μ fragments normal-( x y normal-) μ fragments normal-( x normal-) μ fragments normal-( y normal-) λ fragments normal-( x normal-, y normal-) pmod 2 for-all x normal-, y subscript H k fragments normal-( M normal-; subscript 2 normal-) \mu(x+y)+\mu(x)+\mu(y)\equiv\lambda(x,y)\;\;(\mathop{{\rm mod}}2)\;\forall\,x,% y\in H_{k}(M;\mathbb{Z}_{2})

NTCIR12-MathWiki-15rate: 1

x y w z [ z w ( z x z = y ) ] . fragments for-all x for-all y w for-all z fragments normal-[ z w normal-↔ fragments normal-( z x z y normal-) normal-] normal-. \forall x\,\forall y\,\exists w\,\forall z\,[z\in w\leftrightarrow(z\in xz=y)].

NTCIR12-MathWiki-15rate: 1

x y z ( z y z x ϕ ( z ) ) fragments for-all x y for-all z fragments normal-( z y normal-↔ z x ϕ fragments normal-( z normal-) normal-) \forall x\;\exists y\;\forall z\;(z\in y\leftrightarrow z\in x\wedge\phi(z))

NTCIR12-MathWiki-15rate: 1

w 1 , , w n [ ( x y ϕ ( x , y , w 1 , , w n ) ) A B x A y B ϕ ( x , y , w 1 , , w n ) ] fragments for-all subscript w 1 normal-, normal-… normal-, subscript w n fragments normal-[ fragments normal-( for-all x y ϕ fragments normal-( x normal-, y normal-, subscript w 1 normal-, normal-… normal-, subscript w n normal-) normal-) normal-⇒ for-all A B for-all x A y B ϕ fragments normal-( x normal-, y normal-, subscript w 1 normal-, normal-… normal-, subscript w n normal-) normal-] \forall w_{1},\ldots,w_{n}\,[(\forall x\,\exists\,y\phi(x,y,w_{1},\ldots,w_{n}% ))\Rightarrow\forall A\,\exists B\,\forall x\in A\,\exists y\in B\,\phi(x,y,w_% {1},\ldots,w_{n})]

NTCIR12-MathWiki-15rate: 0

w 1 , , w n A B x A [ y ϕ ( x , y , w 1 , , w n ) y B ϕ ( x , y , w 1 , , w n ) ] fragments for-all subscript w 1 normal-, normal-… normal-, subscript w n for-all A B for-all x A fragments normal-[ y ϕ fragments normal-( x normal-, y normal-, subscript w 1 normal-, normal-… normal-, subscript w n normal-) normal-⇒ y B ϕ fragments normal-( x normal-, y normal-, subscript w 1 normal-, normal-… normal-, subscript w n normal-) normal-] \forall w_{1},\ldots,w_{n}\,\forall A\,\exists B\,\forall x\in A\,[\exists y% \phi(x,y,w_{1},\ldots,w_{n})\Rightarrow\exists y\in B\,\phi(x,y,w_{1},\ldots,w% _{n})]

NTCIR12-MathWiki-15rate: 0

x , y A : x y x y normal-: for-all x y A norm x y norm x norm y \forall x,y\in A:\|x\,y\|\ \leq\|x\|\,\|y\|

NTCIR12-MathWiki-15rate: 1

x , y A , f ( x ) = f ( y ) x = y . formulae-sequence for-all x y A f x f y normal-⇒ x y \forall x,y\in A,f(x)=f(y)\Rightarrow x=y.

NTCIR12-MathWiki-15rate: 0

x , y A , x y f ( x ) f ( y ) . formulae-sequence for-all x y A x y normal-⇒ f x f y \forall x,y\in A,x\neq y\Rightarrow f(x)\neq f(y).

NTCIR12-MathWiki-15rate: 0

x y [ x y z y C ( x ; y z ) ] . fragments for-all x y fragments normal-[ x y normal-→ z y C fragments normal-( x normal-; y z normal-) normal-] normal-. \forall xy\,[x\neq y\rightarrow\exists z\neq y\,C(x;yz)].

NTCIR12-MathWiki-15rate: 0

x , y G for-all x y G \forall x,y\in G

NTCIR12-MathWiki-15rate: 1

A ( [ x A y ϕ ( x , y ) ] B x A y B ϕ ( x , y ) ) fragments for-all A fragments normal-( fragments normal-[ for-all x A y ϕ fragments normal-( x normal-, y normal-) normal-] normal-→ B for-all x A y B ϕ fragments normal-( x normal-, y normal-) normal-) \forall A\;([\forall x\in A\;\exists y\;\phi(x,y)]\to\exists B\;\forall x\in A% \;\exists y\in B\;\phi(x,y))

NTCIR12-MathWiki-15rate: 0

a ( ( x a y ϕ ( x , y ) ) b ( x a y b ϕ ( x , y ) ) ( y b x a ϕ ( x , y ) ) ) fragments for-all a fragments normal-( fragments normal-( for-all x a y ϕ fragments normal-( x normal-, y normal-) normal-) normal-→ b fragments normal-( for-all x a y b ϕ fragments normal-( x normal-, y normal-) normal-) fragments normal-( for-all y b x a ϕ fragments normal-( x normal-, y normal-) normal-) normal-) \forall a((\forall x\in a\;\exists y\;\phi(x,y))\to\exists b\;(\forall x\in a% \;\exists y\in b\;\phi(x,y))\wedge(\forall y\in b\;\exists x\in a\;\phi(x,y)))

NTCIR12-MathWiki-15rate: 0

L ( x ) = i l i x p i , l i GF ( p m ) . formulae-sequence L x subscript i subscript l i superscript x superscript p i subscript l i GF superscript p m L(x)=\sum_{i}l_{i}x^{p^{i}},l_{i}\in\mathrm{GF}(p^{m}).

NTCIR12-MathWiki-15rate: 0

x , y A x y A x,y\in A

NTCIR12-MathWiki-15rate: 0

x ( y ( y x P [ y ] ) P [ x ] ) x P [ x ] fragments for-all x fragments normal-( for-all y fragments normal-( y x normal-→ P fragments normal-[ y normal-] normal-) normal-→ P fragments normal-[ x normal-] normal-) normal-→ for-all x P fragments normal-[ x normal-] \forall x\Big(\forall y(y\in x\rightarrow P[y])\rightarrow P[x]\Big)% \rightarrow\forall x\,P[x]

NTCIR12-MathWiki-15rate: 1

a , b , c X ( b R a c R a b R c ) . fragments for-all a normal-, b normal-, c X fragments normal-( b R a c R a normal-→ b R c normal-) normal-. \forall a,b,c\in X\,(b\,R\,a\land c\,R\,a\to b\,R\,c).

NTCIR12-MathWiki-15rate: 0

x y ( x y G ( x , y ) ) . fragments subscript x subscript for-all y fragments normal-( x y normal-⇒ G fragments normal-( x normal-, y normal-) normal-) normal-. \exists_{x}\forall_{y}(x\neq y\Rightarrow G(x,y)).

NTCIR12-MathWiki-15rate: 1

x y [ z ( z x z y ) z ( x z y z ) ] fragments for-all x for-all y fragments normal-[ for-all z fragments normal-( z x normal-⇔ z y normal-) normal-⇒ for-all z fragments normal-( x z normal-⇔ y z normal-) normal-] \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow\forall z% (x\in z\Leftrightarrow y\in z)]

NTCIR12-MathWiki-15rate: 0

A , B , C , D , P ( C , D ) ( C A D B ) . fragments for-all A normal-, B normal-, for-all C normal-, for-all D normal-, P fragments normal-( C normal-, D normal-) normal-→ fragments normal-( C A normal-→ D B normal-) normal-. \forall A,\exists B,\forall C,\forall D,P(C,D)\rightarrow(C\in A\rightarrow D% \in B).

NTCIR12-MathWiki-15rate: 1

A = { x F ; y F ( x R y y A ) } fragments A fragments normal-{ x F normal-; for-all y F fragments normal-( x R y normal-→ y A normal-) normal-} \Box A=\{x\in F;\,\forall y\in F\,(x\,R\,y\to y\in A)\}

NTCIR12-MathWiki-15rate: 1

x y [ z [ z x z y ] x = y ] . fragments for-all x for-all y fragments normal-[ for-all z fragments normal-[ z x normal-↔ z y normal-] normal-→ x y normal-] normal-. \forall x\forall y[\forall z[z\in x\leftrightarrow z\in y]\rightarrow x=y].

NTCIR12-MathWiki-15rate: 2

z y x [ x y ( x z ϕ ( x ) ) ] . fragments for-all z y for-all x fragments normal-[ x y normal-↔ fragments normal-( x z ϕ fragments normal-( x normal-) normal-) normal-] normal-. \forall z\exists y\forall x[x\in y\leftrightarrow(x\in z\land\phi(x))].

NTCIR12-MathWiki-15rate: 0

x , y A for-all x y A \forall x,y\in A

NTCIR12-MathWiki-15rate: 1

x , y , z ( x F y x F z y = z ) fragments for-all x normal-, y normal-, z fragments normal-( x F y x F z normal-→ y z normal-) \forall x,y,z\,\left(xFy\wedge xFz\to y=z\right)

NTCIR12-MathWiki-15rate: 1

x , y , z ( x R y y R z x R z ) fragments for-all x normal-, y normal-, z fragments normal-( x R y y R z normal-→ x R z normal-) \forall x,y,z\,(xRy\wedge yRz\rightarrow xRz)

NTCIR12-MathWiki-15rate: 2

n ( m [ m > n P ( m ) ] ) fragments for-all n Z fragments normal-( m Z fragments normal-[ m n P fragments normal-( m normal-) normal-] normal-) \forall n\in\mathbb{Z}(\exists m\in\mathbb{Z}[m>n\wedge P(m)])

NTCIR12-MathWiki-15rate: 1

st x st y st t ( t y ( t x ϕ ( t , u 1 , , u n ) ) ) fragments superscript for-all st x superscript st y superscript for-all st t fragments normal-( t y normal-↔ fragments normal-( t x ϕ fragments normal-( t normal-, subscript u 1 normal-, normal-… normal-, subscript u n normal-) normal-) normal-) \forall^{\mathrm{st}}x\,\exists^{\mathrm{st}}y\,\forall^{\mathrm{st}}t\,(t\in y% \leftrightarrow(t\in x\land\phi(t,u_{1},\dots,u_{n})))

NTCIR12-MathWiki-15rate: 0

( g f ) ( x ) := f ( g - 1 x ) x V , g G , f k [ V ] . formulae-sequence assign normal-⋅ g f x f superscript g 1 x formulae-sequence for-all x V formulae-sequence g G f k delimited-[] V (g\cdot f)(x):=f(g^{-1}x)\qquad\forall x\in V,g\in G,f\in k[V].

NTCIR12-MathWiki-15rate: 0

z L Pr x [ y . ϕ ( x , y , z ) ] 2 3 fragments z L subscript Pr x fragments normal-[ y normal-. ϕ fragments normal-( x normal-, y normal-, z normal-) normal-] 2 3 z\in L\implies\Pr\nolimits_{x}[\exists y.\phi(x,y,z)]\geq\tfrac{2}{3}

NTCIR12-MathWiki-15rate: 0

x z M [ x := y ] [ z ] = M [ z ] fragments x z normal-→ M fragments normal-[ x assign y normal-] fragments normal-[ z normal-] M fragments normal-[ z normal-] x\neq z\to M[x:=y][z]=M[z]

NTCIR12-MathWiki-15rate: 0

( z : z F V ( y ) z B V ( b ) ) beta - redex [ λ x . b y ] = b [ x := y ] fragments fragments normal-( for-all z normal-: z F V fragments normal-( y normal-) z B V fragments normal-( b normal-) normal-) normal-→ beta redex fragments normal-[ λ x normal-. b y normal-] b fragments normal-[ x assign y normal-] (\forall z:z\not\in FV(y)z\not\in BV(b))\to\operatorname{beta-redex}[\lambda x% .b\ y]=b[x:=y]

NTCIR12-MathWiki-15rate: 0

x = y and ( L z ) [ x := y ] fragments iff x y and fragments normal-( L z normal-) fragments normal-[ x assign y normal-] \iff x=y\and(L\ z)[x:=y]

NTCIR12-MathWiki-15rate: 0

l x ( y ) = x y , x 𝔤 , y U ( 𝔤 ) formulae-sequence subscript l x y x y formulae-sequence x 𝔤 y U 𝔤 l_{x}(y)=xy,x\in\mathfrak{g},y\in U(\mathfrak{g})

NTCIR12-MathWiki-15rate: 1

ϵ > 0 δ > 0 y X [ d ( x , y ) < δ n 𝐍 d ( f n ( x ) , f n ( y ) ) < ϵ ] . fragments for-all ϵ 0 δ 0 for-all y X fragments normal-[ d fragments normal-( x normal-, y normal-) δ normal-⇒ for-all n N d fragments normal-( superscript f n fragments normal-( x normal-) normal-, superscript f n fragments normal-( y normal-) normal-) ϵ normal-] normal-. \forall\epsilon>0\ \exists\delta>0\ \forall y\in X\ \left[d(x,y)<\delta% \Rightarrow\forall n\in\mathbf{N}\ d\left(f^{n}(x),f^{n}(y)\right)<\epsilon% \right].

NTCIR12-MathWiki-15rate: 1

X U p [ C U M p ( X ) x , y [ X ( x ) X ( y ) ¬ ( x = y ) ] x , y [ X ( x ) X ( y ) X ( x y ) ] ] fragments for-all X subscript U p fragments normal-[ C U subscript M p fragments normal-( X normal-) normal-⇔ x normal-, y fragments normal-[ X fragments normal-( x normal-) X fragments normal-( y normal-) fragments normal-( x y normal-) normal-] for-all x normal-, y fragments normal-[ X fragments normal-( x normal-) X fragments normal-( y normal-) normal-⇒ X fragments normal-( x direct-sum y normal-) normal-] normal-] \forall X\subseteq U_{p}[CUM_{p}(X)\Leftrightarrow\exists x,y[X(x)\,\wedge\,X(% y)\,\wedge\,\neg(x=y)]\;\wedge\;\forall x,y[X(x)\,\wedge\,X(y)\Rightarrow X(x% \,\oplus\,y)]]

NTCIR12-MathWiki-15rate: 1

P x y z [ O z x O z y ] . fragments P x y normal-↔ for-all z fragments normal-[ O z x normal-→ O z y normal-] normal-. Pxy\leftrightarrow\forall z[Ozx\rightarrow Ozy].

NTCIR12-MathWiki-15rate: 1

x \exist y [ P y x and ( C z y O z x ) and ¬ ( P x y and ( C z x O z y ) ) ] . fragments for-all x \exist y fragments normal-[ P y x and fragments normal-( C z y normal-→ O z x normal-) and fragments normal-( P x y and fragments normal-( C z x normal-→ O z y normal-) normal-) normal-] normal-. \forall x\exist y[Pyx\and(Czy\rightarrow Ozx)\and\lnot(Pxy\and(Czx\rightarrow Ozy% ))].

NTCIR12-MathWiki-15rate: 1

x , y . P ( x , y ) Q ( f ( x ) ) formulae-sequence for-all x y P x y Q f x \forall x,y.P(x,y)\vee Q(f(x))

NTCIR12-MathWiki-15rate: 0

C := { b B : p ( x ) A [ x ] , which is monic and such that p ( b ) = 0 } . assign C conditional-set b B formulae-sequence p x A delimited-[] x which is monic and such that p b 0 C:=\{b\in B:\exists\,p(x)\in A[x]\,,\hbox{ which is monic and such that }p(b)=% 0\}\,.

NTCIR12-MathWiki-15rate: 1

a s [ ( M a and x [ x s y ( x y and y a ) ] ) M s ] . fragments for-all a for-all s fragments normal-[ fragments normal-( M a and for-all x fragments normal-[ x s normal-↔ y fragments normal-( x y and y a normal-) normal-] normal-) normal-→ M s normal-] normal-. \forall a\,\forall s\,[(Ma\and\forall x\,[x\in s\leftrightarrow\exists y\,(x% \in y\and y\in a)])\rightarrow Ms].

NTCIR12-MathWiki-15rate: 1

y [ M y and y and z ( z y x [ x y and w ( w x [ w = z w z ] ) ] ) ] . fragments y fragments normal-[ M y and y and for-all z fragments normal-( z y normal-→ x fragments normal-[ x y and for-all w fragments normal-( w x normal-↔ fragments normal-[ w z w z normal-] normal-) normal-] normal-) normal-] normal-. \exists y[My\and\varnothing\in y\and\forall z(z\in y\rightarrow\exists x[x\in y% \and\forall w(w\in x\leftrightarrow[w=zw\in z])])].

NTCIR12-MathWiki-15rate: 0

x , y A x y A x,y\in A

NTCIR12-MathWiki-15rate: 0

x , y A x y x y formulae-sequence for-all x y A norm x y norm x norm y \forall x,y\in A\qquad\|xy\|\leq\|x\|\|y\|

NTCIR12-MathWiki-15rate: 1

( x ) E q ( x , x ) ( x , y , z ) [ E q ( x , y ) ( E q ( x , z ) E q ( y , z ) ) ] fragments fragments normal-( for-all x normal-) E q fragments normal-( x normal-, x normal-) fragments normal-( for-all x normal-, y normal-, z normal-) fragments normal-[ E q fragments normal-( x normal-, y normal-) normal-→ fragments normal-( E q fragments normal-( x normal-, z normal-) normal-→ E q fragments normal-( y normal-, z normal-) normal-) normal-] (\forall x)Eq(x,x)\wedge(\forall x,y,z)[Eq(x,y)\rightarrow(Eq(x,z)\rightarrow Eq% (y,z))]

NTCIR12-MathWiki-15rate: 0

x , y N for-all x y N \forall x,y\in N

NTCIR12-MathWiki-15rate: 0

x , y N for-all x y N \forall x,y\in N

NTCIR12-MathWiki-15rate: 1

x , y N x < y z N x + z = y for-all x y N x y normal-⇒ z N x z y \forall x,y\in Nx<y\Rightarrow\exists z\in Nx+z=y

NTCIR12-MathWiki-15rate: 0

x , y N for-all x y N \forall x,y\in N

NTCIR12-MathWiki-15rate: 0

x , y : ¬ ( x < y y < x ) normal-: for-all x y x y y x \forall x,y:\neg\;(x<y\;\wedge\;y<x)

NTCIR12-MathWiki-15rate: 0

x , y A x y A x,y\in A

NTCIR12-MathWiki-15rate: 0

g p M ( u , v ) = g f ( p ) N ( d f ( u ) , d f ( v ) ) p M , u , v T p M . formulae-sequence subscript superscript g M p u v subscript superscript g N f p d f u d f v formulae-sequence for-all p M for-all u v subscript T p M g^{M}_{p}(u,v)=g^{N}_{f(p)}(df(u),df(v))\qquad\forall p\in M,\forall u,v\in T_% {p}M.

NTCIR12-MathWiki-15rate: 0

y x ( x y P ( x ) ) fragments y for-all x fragments normal-( x y iff P fragments normal-( x normal-) normal-) \exists y\forall x(x\in y\iff P(x))

NTCIR12-MathWiki-15rate: 1

x [ y ( R x y z [ R y z ] ) y ( R x y z [ R y z z = y ] ) ] fragments for-all x fragments normal-[ for-all y fragments normal-( R x y normal-→ z fragments normal-[ R y z normal-] normal-) normal-→ y fragments normal-( R x y for-all z fragments normal-[ R y z normal-→ z y normal-] normal-) normal-] \forall x[\forall y(Rxy\rightarrow\exists z[Ryz])\rightarrow\exists y(Rxy% \wedge\forall z[Ryz\rightarrow z=y])]

NTCIR12-MathWiki-15rate: 0

x , y A x y A x,y\in A

NTCIR12-MathWiki-15rate: 4

x , y A [ x y ¬ z X [ z x z y ] ] . fragments for-all x normal-, y A fragments normal-[ x y normal-→ z X fragments normal-[ z x z y normal-] normal-] normal-. \forall x,y\in A\;[x\neq y\rightarrow\neg\exists z\in X\;[z\leq x\land z\leq y% ]].

NTCIR12-MathWiki-15rate: 1

x , y [ x y [ x R y y R x ] ] fragments for-all x normal-, y fragments normal-[ x y normal-→ fragments normal-[ x R y y R x normal-] normal-] \forall x,y[x\neq y\rightarrow[xRy\vee yRx]]

NTCIR12-MathWiki-15rate: 0

x , y A , f ( x + y ) f ( x ) + f ( y ) . formulae-sequence for-all x y A f x y f x f y \forall x,y\in A,f(x+y)\leq f(x)+f(y).

NTCIR12-MathWiki-15rate: 0

x , y G for-all x y G \forall x,y\in G

NTCIR12-MathWiki-15rate: 0

u S v x , y M ( x u y S x v y S ) . fragments u subscript S v normal-⇔ for-all x normal-, y M fragments normal-( x u y S normal-⇔ x v y S normal-) normal-. u\equiv_{S}v\Leftrightarrow\forall x,y\in M(xuy\in S\Leftrightarrow xvy\in S).

NTCIR12-MathWiki-15rate: 1

r x [ F x r [ y ( y x B y r ) and ¬ B x r ] ] . fragments for-all r for-all x fragments normal-[ F x r normal-↔ fragments normal-[ for-all y fragments normal-( y x normal-→ B y r normal-) and B x r normal-] normal-] normal-. \forall r\forall x[Fxr\leftrightarrow[\forall y(y\in x\rightarrow Byr)\and% \lnot Bxr]]\,.

NTCIR12-MathWiki-15rate: 0

\exist r y [ A ( y ) B y r ] \exist x y [ y x A ( y ) ] . fragments \exist r for-all y fragments normal-[ A fragments normal-( y normal-) normal-→ B y r normal-] normal-→ \exist x for-all y fragments normal-[ y x normal-↔ A fragments normal-( y normal-) normal-] normal-. \exist r\forall y[A(y)\rightarrow Byr]\rightarrow\exist x\forall y[y\in x% \leftrightarrow A(y)]\,.

NTCIR12-MathWiki-15rate: 0

x y [ x y z y ( 𝒫 ( z ) y 𝒫 ( z ) y ) z 𝒫 ( y ) ( ¬ z y z y ) ] fragments for-all x y fragments normal-[ x y for-all z y fragments normal-( P fragments normal-( z normal-) y P fragments normal-( z normal-) y normal-) for-all z P fragments normal-( y normal-) fragments normal-( z y normal-→ z y normal-) normal-] \forall x\exists y[x\in y\wedge\forall z\in y(\mathcal{P}(z)\subseteq y\wedge% \mathcal{P}(z)\in y)\wedge\forall z\in\mathcal{P}(y)(\neg z\approx y\to z\in y)]

NTCIR12-MathWiki-15rate: 0

x , y A 1 x y subscript A 1 x,y\in A_{1}

NTCIR12-MathWiki-15rate: 1

n A x [ x A iff x n ] . fragments for-all n N A N for-all x N fragments normal-[ x A iff x n normal-] normal-. \forall n\in\mathbb{N}\ \exists A\subseteq\mathbb{N}\ \forall x\in\mathbb{N}\ % [x\in A\,\text{ iff }x\leq n].

NTCIR12-MathWiki-15rate: 0

T x y 2 = x T * y 1 , x D ( T ) . formulae-sequence subscript inner-product T x y 2 subscript inner-product x superscript T y 1 x D T \langle Tx\mid y\rangle_{2}=\left\langle x\mid T^{*}y\right\rangle_{1},\qquad x% \in D(T).

NTCIR12-MathWiki-15rate: 0

x , y U , x y formulae-sequence for-all x y U x y \forall x,y\in U,~{}x\neq y

NTCIR12-MathWiki-15rate: 0

x , y U , x y formulae-sequence for-all x y U x y \forall x,y\in U,~{}x\neq y

NTCIR12-MathWiki-15rate: 0

x , y U , x y formulae-sequence for-all x y U x y \forall x,y\in U,~{}x\neq y

NTCIR12-MathWiki-15rate: 0

x y z w [ w z ( w = x w = y ) ] . fragments for-all x for-all y z for-all w fragments normal-[ w z normal-↔ fragments normal-( w x w y normal-) normal-] normal-. \forall x\forall y\exists z\forall w[w\in z\leftrightarrow(w=xw=y)].

NTCIR12-MathWiki-15rate: 0

x , y X , x y x S y S formulae-sequence for-all x y X x y x S normal-⇒ y S \forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S

NTCIR12-MathWiki-15rate: 0

x < y z [ x < z < y ] . fragments x y normal-→ z fragments normal-[ x z y normal-] normal-. x<y\rightarrow\exists z[x<z<y].

NTCIR12-MathWiki-15rate: 0

z [ z < x z < y ] x y . fragments for-all z fragments normal-[ z x normal-→ z y normal-] normal-→ x y normal-. \forall z[z<x\rightarrow z<y]\rightarrow x\leq y.

NTCIR12-MathWiki-15rate: 0

x y z ( x z y z ) . fragments for-all x for-all y z fragments normal-( x z y z normal-) normal-. \forall x\forall y\exists z(x\in z\land y\in z).

NTCIR12-MathWiki-15rate: 1

x y z [ z x z y ] . fragments for-all x y for-all z fragments normal-[ z x normal-⇒ z y normal-] normal-. \forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y].

NTCIR12-MathWiki-16rate: 3

λ piv = e ( λ ) λ d λ e ( λ ) λ - 1 d λ λ piv e λ λ d λ e λ superscript λ 1 d λ \lambda\text{piv}=\sqrt{\frac{\int e(\lambda)\lambda d\lambda}{\int e(\lambda)% \lambda^{-1}d\lambda}}

NTCIR12-MathWiki-16rate: 0

a ( t ) = e 0 t δ u d u a t superscript e superscript subscript 0 t subscript δ u d u a(t)=e^{\int_{0}^{t}\delta_{u}\,du}

NTCIR12-MathWiki-16rate: 1

σ y 2 ( τ ) = 0 S y ( f ) 2 sin 4 π τ f ( π τ f ) 2 d f superscript subscript σ y 2 τ superscript subscript 0 subscript S y f 2 superscript 4 π τ f superscript π τ f 2 d f \sigma_{y}^{2}(\tau)=\int_{0}^{\infty}S_{y}(f)\frac{2\sin^{4}\pi\tau f}{(\pi% \tau f)^{2}}\,df

NTCIR12-MathWiki-16rate: 0

x ( t ) = 1 2 a 0 ( t ) + n = 1 [ a n ( t ) cos ( 2 π n 0 t f 0 ( τ ) d τ ) - b n ( t ) sin ( 2 π n 0 t f 0 ( τ ) d τ ) ] x t 1 2 subscript a 0 t superscript subscript n 1 delimited-[] subscript a n t 2 π n superscript subscript 0 t subscript f 0 τ d τ subscript b n t 2 π n superscript subscript 0 t subscript f 0 τ d τ x(t)=\frac{1}{2}a_{0}(t)\ +\ \sum_{n=1}^{\infty}\left[a_{n}(t)\cos\left(2\pi n% \int_{0}^{t}f_{0}(\tau)\,d\tau\right)-b_{n}(t)\sin\left(2\pi n\int_{0}^{t}f_{0% }(\tau)\,d\tau\right)\right]

NTCIR12-MathWiki-16rate: 0

T = e - 0 μ ( z ) d z = 10 - 0 μ 10 ( z ) d z , T superscript e superscript subscript 0 normal-ℓ μ z normal-d z superscript 10 superscript subscript 0 normal-ℓ subscript μ 10 z normal-d z T=e^{-\int_{0}^{\ell}\mu(z)\mathrm{d}z}=10^{-\int_{0}^{\ell}\mu_{10}(z)\mathrm% {d}z},

NTCIR12-MathWiki-16rate: 0

M = 1 1 - 0 L α ( x ) d x M 1 1 superscript subscript 0 L α x d x M=\frac{1}{1-\int_{0}^{L}\alpha(x)\,dx}

NTCIR12-MathWiki-16rate: 4

D 4 σ = 4 σ = 4 - - I ( x , y ) ( x - x ¯ ) 2 d x d y - - I ( x , y ) d x d y D 4 σ 4 σ 4 superscript subscript superscript subscript I x y superscript x normal-¯ x 2 d x d y superscript subscript superscript subscript I x y d x d y D4\sigma=4\sigma=4\sqrt{\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(% x,y)(x-\bar{x})^{2}\,dx\,dy}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x% ,y)\,dx\,dy}}

NTCIR12-MathWiki-16rate: 0

T = e - 0 μ ( z ) d z = 10 - 0 μ 10 ( z ) d z , T superscript e superscript subscript 0 normal-ℓ μ z normal-d z superscript 10 superscript subscript 0 normal-ℓ subscript μ 10 z normal-d z T=e^{-\int_{0}^{\ell}\mu(z)\mathrm{d}z}=10^{-\int_{0}^{\ell}\mu_{10}(z)\mathrm% {d}z},

NTCIR12-MathWiki-16rate: 0

J α ( x ) = 1 π 0 π cos ( α τ - x sin τ ) d τ - sin ( α π ) π 0 e - x sinh ( t ) - α t d t . subscript J α x 1 π superscript subscript 0 π α τ x τ d τ α π π superscript subscript 0 superscript e x t α t d t J_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{\pi}\cos(\alpha\tau-x\sin\tau)\,d\tau-% \frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}e^{-x\sinh(t)-\alpha t}\,dt.

NTCIR12-MathWiki-16rate: 1

p ( t ) = 0 h ( τ ) q ( t - τ ) d τ p t superscript subscript 0 h τ q t τ d τ p(t)=\int_{0}^{\infty}h(\tau)q(t-\tau)\,d\tau

NTCIR12-MathWiki-16rate: 0

[ ( τ s - τ ¯ s ) 2 ] ¯ 1 2 = 1.4 s , superscript normal-¯ delimited-[] superscript subscript τ s subscript normal-¯ τ s 2 1 2 1.4 s \overline{\left[\left(\tau_{s}-\bar{\tau}_{s}\right)^{2}\right]}^{\frac{1}{2}}% =1.4\sqrt{s},

NTCIR12-MathWiki-16rate: 1

= ( y i - m x i - b ) 2 n - 2 absent superscript subscript y i m subscript x i b 2 n 2 =\sqrt{\frac{\sum{(y_{i}-mx_{i}-b)}^{2}}{n-2}}

NTCIR12-MathWiki-16rate: 1

f ( t ) = 0 h ( τ ) s ( t - τ ) d τ f t superscript subscript 0 h τ s t τ d τ f(t)=\int_{0}^{\infty}h(\tau)s(t-\tau)\,d\tau

NTCIR12-MathWiki-16rate: 2

C = x g ( x ) d x g ( x ) d x C x g x d x g x d x C=\frac{\int xg(x)\;dx}{\int g(x)\;dx}

NTCIR12-MathWiki-16rate: 2

C k = z S k ( z ) d z S k ( z ) d z subscript C k z subscript S k z d z subscript S k z d z C_{k}=\frac{\int zS_{k}(z)\;dz}{\int S_{k}(z)\;dz}

NTCIR12-MathWiki-16rate: 0

lim λ 0 λ ( 1 - x λ ) α f ( x ) d x subscript normal-→ λ superscript subscript 0 λ superscript 1 x λ α f x d x \lim_{\lambda\to\infty}\int_{0}^{\lambda}\left(1-\frac{x}{\lambda}\right)^{% \alpha}f(x)\,dx

NTCIR12-MathWiki-16rate: 2

s p = p ( 1 - p ) n subscript s p p 1 p n s_{p}=\sqrt{\frac{p\,(1-p)}{n}}

NTCIR12-MathWiki-16rate: 0

a ( n ) = e 0 n δ t d t , a n superscript e superscript subscript 0 n subscript δ t d t a(n)=e^{\int_{0}^{n}\delta_{t}\,dt}\ ,

NTCIR12-MathWiki-16rate: 2

= def - f ( τ ) g ( t - τ ) d τ superscript def absent superscript subscript f τ g t τ d τ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}f(\tau)\,g(t-\tau)\,d\tau

NTCIR12-MathWiki-16rate: 2

= - f ( t - τ ) g ( τ ) d τ . absent superscript subscript f t τ g τ d τ =\int_{-\infty}^{\infty}f(t-\tau)\,g(\tau)\,d\tau.

NTCIR12-MathWiki-16rate: 4

τ rms = 0 ( τ - τ ¯ ) 2 A c ( τ ) d τ 0 A c ( τ ) d τ subscript τ rms superscript subscript 0 superscript τ normal-¯ τ 2 subscript A c τ d τ superscript subscript 0 subscript A c τ d τ \tau_{\,\text{rms}}=\sqrt{\frac{\int_{0}^{\infty}(\tau-\overline{\tau})^{2}A_{% c}(\tau)d\tau}{\int_{0}^{\infty}A_{c}(\tau)d\tau}}

NTCIR12-MathWiki-16rate: 0

= - f ( τ ) δ ( τ - ( t - T ) ) d τ absent superscript subscript f τ δ τ t T d τ =\int\limits_{-\infty}^{\infty}f(\tau)\delta(\tau-(t-T))\,d\tau

NTCIR12-MathWiki-16rate: 1

v C ( t ) = V 0 + 1 C 0 t i C ( τ ) d τ i L ( t ) = I 0 + 1 L 0 t v L ( τ ) d τ iff subscript v C t subscript V 0 1 C superscript subscript 0 t subscript i C τ d τ subscript i L t subscript I 0 1 L superscript subscript 0 t subscript v L τ d τ v_{C}(t)=V_{0}+{1\over C}\int_{0}^{t}i_{C}(\tau)\,d\tau\iff i_{L}(t)=I_{0}+{1% \over L}\int_{0}^{t}v_{L}(\tau)\,d\tau

NTCIR12-MathWiki-16rate: 2

= 0 G ( τ ) F ( t - τ ) d τ absent superscript subscript 0 G τ F t τ d τ =\int_{0}^{\infty}G(\tau)F(t-\tau)\,d\tau

NTCIR12-MathWiki-16rate: 0

= - t G ( t - τ ) F ( τ ) d τ absent superscript subscript t G t τ F τ d τ =\int_{-\infty}^{t}G(t-\tau)F(\tau)\,d\tau

NTCIR12-MathWiki-16rate: 1

( A t - A ¯ ) 2 ¯ lim τ 1 τ 0 τ ( A t - A ¯ ) 2 d t , normal-¯ superscript subscript A t normal-¯ A 2 subscript normal-→ τ 1 τ superscript subscript 0 τ superscript subscript A t normal-¯ A 2 d t \overline{\left(A_{t}-\overline{A}\right)^{2}}\equiv\lim_{\tau\to\infty}\frac{% 1}{\tau}\int_{0}^{\tau}\left(A_{t}-\overline{A}\right)^{2}dt,

NTCIR12-MathWiki-16rate: 0

x = 0 L cos θ d s = 0 L cos [ ( a s ) 2 ] d s x absent superscript subscript 0 L θ d s missing-subexpression absent superscript subscript 0 L superscript a s 2 d s \begin{aligned}\displaystyle x&\displaystyle=\int_{0}^{L}\cos\theta\,ds\\ &\displaystyle=\int_{0}^{L}\cos\left[(as)^{2}\right]ds\end{aligned}

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Q = 0 t I ( τ ) d τ Q superscript subscript 0 t I τ d τ Q=\int_{0}^{t}I(\tau)\ d\tau

NTCIR12-MathWiki-16rate: 1

Δ ( x ) = 0 d τ D X e - 0 τ ( x ˙ 2 / 2 + m 2 ) d τ normal-Δ x superscript subscript 0 d τ D X superscript e superscript subscript 0 τ superscript normal-˙ x 2 2 superscript m 2 d superscript τ normal-′ \Delta(x)=\int_{0}^{\infty}d\tau\int DXe^{-\int_{0}^{\tau}(\dot{x}^{2}/2+m^{2}% )d\tau^{\prime}}

NTCIR12-MathWiki-16rate: 0

e - 0 t V ( x ( τ ) ) d τ superscript e superscript subscript 0 t V x τ d τ e^{-\int_{0}^{t}V(x(\tau))\,d\tau}

NTCIR12-MathWiki-16rate: 1

f ( t ) = 2 0 - f ( τ ) cos 2 π λ ( τ - t ) d τ d λ . f t 2 superscript subscript 0 superscript subscript f τ 2 π λ τ t d τ d λ f(t)=2\int_{0}^{\infty}\int_{-\infty}^{\infty}f(\tau)\cos 2\pi\lambda(\tau-t)d% \tau d\lambda.

NTCIR12-MathWiki-16rate: 1

( J α ) ( J β f ) ( x ) = 1 Γ ( α ) 0 x ( x - t ) α - 1 ( J β f ) ( t ) d t = 1 Γ ( α ) Γ ( β ) 0 x 0 t ( x - t ) α - 1 ( t - s ) β - 1 f ( s ) d s d t = 1 Γ ( α ) Γ ( β ) 0 x f ( s ) ( s x ( x - t ) α - 1 ( t - s ) β - 1 d t ) d s superscript J α superscript J β f x absent 1 normal-Γ α superscript subscript 0 x superscript x t α 1 superscript J β f t d t missing-subexpression absent 1 normal-Γ α normal-Γ β superscript subscript 0 x superscript subscript 0 t superscript x t α 1 superscript t s β 1 f s d s d t missing-subexpression absent 1 normal-Γ α normal-Γ β superscript subscript 0 x f s superscript subscript s x superscript x t α 1 superscript t s β 1 d t d s \begin{aligned}\displaystyle(J^{\alpha})(J^{\beta}f)(x)&\displaystyle=\frac{1}% {\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha-1}(J^{\beta}f)(t)\;dt\\ &\displaystyle=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{x}\int_{0}^{t}(x% -t)^{\alpha-1}(t-s)^{\beta-1}f(s)\;ds\;dt\\ &\displaystyle=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{0}^{x}f(s)\left(\int% _{s}^{x}(x-t)^{\alpha-1}(t-s)^{\beta-1}\;dt\right)ds\end{aligned}

NTCIR12-MathWiki-16rate: 0

= A c cos ( 2 π f c t + 2 π f Δ 0 t x m ( τ ) d τ ) absent subscript A c 2 π subscript f c t 2 π subscript f normal-Δ superscript subscript 0 t subscript x m τ d τ =A_{c}\cos\left(2\pi f_{c}t+2\pi f_{\Delta}\int_{0}^{t}x_{m}(\tau)d\tau\right)

NTCIR12-MathWiki-16rate: 1

0 t x m ( τ ) d τ = A m cos ( 2 π f m t ) 2 π f m superscript subscript 0 t subscript x m τ d τ subscript A m 2 π subscript f m t 2 π subscript f m \int_{0}^{t}x_{m}(\tau)d\tau=\frac{A_{m}\cos(2\pi f_{m}t)}{2\pi f_{m}}\,

NTCIR12-MathWiki-16rate: 4

( Δ x ) 2 = - ( x - μ ) 2 f ( x ) f * ( x ) d x - f ( x ) f * ( x ) d x superscript normal-Δ x 2 superscript subscript superscript x μ 2 f x superscript f x d x superscript subscript f x superscript f x d x (\Delta x)^{2}=\frac{\int_{-\infty}^{\infty}(x-\mu)^{2}f(x)f^{*}(x)\,dx}{\int_% {-\infty}^{\infty}f(x)f^{*}(x)\,dx}

NTCIR12-MathWiki-16rate: 4

( Δ k ) 2 = - ( k - k 0 ) 2 F ( k ) F * ( k ) d k - F ( k ) F * ( k ) d k superscript normal-Δ k 2 superscript subscript superscript k subscript k 0 2 F k superscript F k d k superscript subscript F k superscript F k d k (\Delta k)^{2}=\frac{\int_{-\infty}^{\infty}(k-k_{0})^{2}F(k)F^{*}(k)\,dk}{% \int_{-\infty}^{\infty}F(k)F^{*}(k)\,dk}

NTCIR12-MathWiki-16rate: 1

G E H = 2 ( M - C ) 2 M + C G E H 2 superscript M C 2 M C GEH=\sqrt{\frac{2(M-C)^{2}}{M+C}}

NTCIR12-MathWiki-16rate: 4

T = x = 0 x ( x ) m ( x ) d x x = 0 ( x ) m ( x ) d x T superscript subscript x 0 x normal-ℓ x m x normal-d x superscript subscript x 0 normal-ℓ x m x normal-d x T=\frac{\int_{x=0}^{\infty}x\ell(x)m(x)\,\mathrm{d}x}{\int_{x=0}^{\infty}\ell(% x)m(x)\,\mathrm{d}x}

NTCIR12-MathWiki-16rate: 2

G W P ( x ) = 0 T H a x [ x ( t ) ] d t 0 T H a r [ r ( t ) ] d t G W P x superscript subscript 0 T H normal-⋅ subscript a x delimited-[] x t d t superscript subscript 0 T H normal-⋅ subscript a r delimited-[] r t d t GWP\left(x\right)=\frac{\int_{0}^{TH}a_{x}\cdot\left[x(t)\right]dt}{\int_{0}^{% TH}a_{r}\cdot\left[r(t)\right]dt}

NTCIR12-MathWiki-16rate: 1

q x x = ( x - x ¯ ) 2 I ( x , y ) I ( x , y ) subscript q x x superscript x normal-¯ x 2 I x y I x y q_{xx}=\frac{\sum(x-\bar{x})^{2}I(x,y)}{\sum I(x,y)}

NTCIR12-MathWiki-16rate: 1

Y ( t ) = χ i X ( t ) + 0 Φ d ( τ ) X ( t - τ ) d τ , Y t χ i X t superscript subscript 0 normal-Φ d τ X t τ normal-d τ Y(t)=\chi\text{i}X(t)+\int_{0}^{\infty}\Phi\text{d}(\tau)X(t-\tau)\,\mathrm{d}\tau,

NTCIR12-MathWiki-16rate: 0

D γ ( γ ( b ) | | γ ( a ) ) = a b ( b - s ) g γ ( s ) d s fragments subscript D γ fragments normal-( γ fragments normal-( b normal-) normal-| normal-| γ fragments normal-( a normal-) normal-) superscript subscript a b fragments normal-( b s normal-) subscript g γ fragments normal-( s normal-) d s D_{\gamma}(\gamma(b)||\gamma(a))=\int_{a}^{b}(b-s)g_{\gamma}(s)ds

NTCIR12-MathWiki-16rate: 3

δ ( x ) = - θ f ( x 1 - θ , , x n - θ ) d θ - f ( x 1 - θ , , x n - θ ) d θ . δ x superscript subscript θ f subscript x 1 θ normal-… subscript x n θ d θ superscript subscript f subscript x 1 θ normal-… subscript x n θ d θ \delta(x)=\frac{\int_{-\infty}^{\infty}{\theta f(x_{1}-\theta,\dots,x_{n}-% \theta)d\theta}}{\int_{-\infty}^{\infty}{f(x_{1}-\theta,\dots,x_{n}-\theta)d% \theta}}.

NTCIR12-MathWiki-16rate: 3

M ananiso = M s 0 π e 0.5 ( E ( 1 ) + E ( 2 ) ) sin ( θ ) cos ( θ ) d θ 0 π e 0.5 ( E ( 1 ) + E ( 2 ) ) sin ( θ ) d θ M ananiso M s superscript subscript 0 π superscript e 0.5 E 1 E 2 θ θ d θ superscript subscript 0 π superscript e 0.5 E 1 E 2 θ d θ M\text{an}\text{aniso}=M\text{s}\frac{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(% \theta)\cos(\theta)\,d\theta}{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(\theta)\,% d\theta}

NTCIR12-MathWiki-16rate: 1

𝐄 [ G | H ] = 0 T k ( t ) 𝐄 [ x ( t ) | H ] d t = 0 𝐄 [ G | K ] = 0 T k ( t ) 𝐄 [ x ( t ) | K ] d t = 0 T k ( t ) S ( t ) d t ρ 𝐄 [ G 2 | H ] = 0 T 0 T k ( t ) k ( s ) R N ( t , s ) d t d s = 0 T k ( t ) ( 0 T k ( s ) R N ( t , s ) d s ) = 0 T k ( t ) S ( t ) d t = ρ Var [ G | H ] = 𝐄 [ G 2 | H ] - ( 𝐄 [ G | H ] ) 2 = ρ 𝐄 [ G 2 | K ] = 0 T 0 T k ( t ) k ( s ) 𝐄 [ x ( t ) x ( s ) ] d t d s = 0 T 0 T k ( t ) k ( s ) ( R N ( t , s ) + S ( t ) S ( s ) ) d t d s = ρ + ρ 2 Var [ G | K ] = 𝐄 [ G 2 | K ] - ( 𝐄 [ G | K ] ) 2 = ρ + ρ 2 - ρ 2 = ρ fragments E fragments normal-[ G normal-| H normal-] fragments subscript superscript T 0 k fragments normal-( t normal-) E fragments normal-[ x fragments normal-( t normal-) normal-| H normal-] d t 0 fragments E fragments normal-[ G normal-| K normal-] fragments subscript superscript T 0 k fragments normal-( t normal-) E fragments normal-[ x fragments normal-( t normal-) normal-| K normal-] d t subscript superscript T 0 k fragments normal-( t normal-) S fragments normal-( t normal-) d t ρ fragments E fragments normal-[ superscript G 2 normal-| H normal-] absent subscript superscript T 0 subscript superscript T 0 k t k s subscript R N t s d t d s subscript superscript T 0 k t subscript superscript T 0 k s subscript R N t s d s subscript superscript T 0 k t S t d t ρ fragments Var fragments normal-[ G normal-| H normal-] fragments E fragments normal-[ superscript G 2 normal-| H normal-] superscript fragments normal-( E fragments normal-[ G normal-| H normal-] normal-) 2 ρ fragments E fragments normal-[ superscript G 2 normal-| K normal-] absent subscript superscript T 0 subscript superscript T 0 k t k s 𝐄 delimited-[] x t x s d t d s subscript superscript T 0 subscript superscript T 0 k t k s subscript R N t s S t S s d t d s ρ superscript ρ 2 fragments Var fragments normal-[ G normal-| K normal-] fragments E fragments normal-[ superscript G 2 normal-| K normal-] superscript fragments normal-( E fragments normal-[ G normal-| K normal-] normal-) 2 ρ superscript ρ 2 superscript ρ 2 ρ \begin{aligned}\displaystyle\mathbf{E}[G|H]&\displaystyle=\int^{T}_{0}k(t)% \mathbf{E}[x(t)|H]dt=0\\ \displaystyle\mathbf{E}[G|K]&\displaystyle=\int^{T}_{0}k(t)\mathbf{E}[x(t)|K]% dt=\int^{T}_{0}k(t)S(t)dt\equiv\rho\\ \displaystyle\mathbf{E}[G^{2}|H]&\displaystyle=\int^{T}_{0}\int^{T}_{0}k(t)k(s% )R_{N}(t,s)dtds=\int^{T}_{0}k(t)\left(\int^{T}_{0}k(s)R_{N}(t,s)ds\right)=\int% ^{T}_{0}k(t)S(t)dt=\rho\\ \displaystyle\,\text{Var}[G|H]&\displaystyle=\mathbf{E}[G^{2}|H]-(\mathbf{E}[G% |H])^{2}=\rho\\ \displaystyle\mathbf{E}[G^{2}|K]&\displaystyle=\int^{T}_{0}\int^{T}_{0}k(t)k(s% )\mathbf{E}[x(t)x(s)]dtds=\int^{T}_{0}\int^{T}_{0}k(t)k(s)(R_{N}(t,s)+S(t)S(s)% )dtds=\rho+\rho^{2}\\ \displaystyle\,\text{Var}[G|K]&\displaystyle=\mathbf{E}[G^{2}|K]-(\mathbf{E}[G% |K])^{2}=\rho+\rho^{2}-\rho^{2}=\rho\end{aligned}

NTCIR12-MathWiki-16rate: 1

d P d t = 0 t 𝐀 ( t - τ ) P ( τ ) d τ d normal-→ P d t subscript superscript t 0 𝐀 t τ normal-→ P τ d τ \frac{d\vec{P}}{dt}=\int^{t}_{0}\mathbf{A}(t-\tau)\vec{P}(\tau)d\tau

NTCIR12-MathWiki-16rate: 3

α sun = 0 α λ I λ sun ( λ ) d λ 0 I λ sun ( λ ) d λ subscript α sun superscript subscript 0 subscript α λ subscript I λ sun λ d λ superscript subscript 0 subscript I λ sun λ d λ \alpha_{\mathrm{sun}}=\displaystyle\frac{\int_{0}^{\infty}\alpha_{\lambda}I_{% \lambda\mathrm{sun}}(\lambda)\,d\lambda}{\int_{0}^{\infty}I_{\lambda\mathrm{% sun}}(\lambda)\,d\lambda}

NTCIR12-MathWiki-16rate: 2

v e = v e 0 f ( v x ) v x d v x - f ( v x ) d v x delimited-⟨⟩ subscript v e superscript subscript subscript v e 0 f subscript v x subscript v x d subscript v x superscript subscript f subscript v x d subscript v x \langle v_{e}\rangle=\frac{\int_{v_{e0}}^{\infty}f(v_{x})\,v_{x}\,dv_{x}}{\int% _{-\infty}^{\infty}f(v_{x})\,dv_{x}}

NTCIR12-MathWiki-16rate: 2

y ( t ) = 0 T x ( t - τ ) h ( τ ) d τ y t superscript subscript 0 T x t τ h τ d τ y(t)=\int_{0}^{T}x(t-\tau)\,h(\tau)\,d\tau

NTCIR12-MathWiki-16rate: 1

0 t ROI ( τ ) d τ = ( - 𝐔 n T 𝐊 - 1 𝐐 + V p ) 0 t C p ( τ ) d τ + 𝐔 n T 𝐊 - 1 𝐀 superscript subscript 0 t ROI τ d τ superscript subscript 𝐔 n T superscript 𝐊 1 𝐐 subscript V p superscript subscript 0 t subscript C p τ d τ superscript subscript 𝐔 n T superscript 𝐊 1 𝐀 \int_{0}^{t}\mathrm{ROI}(\tau)\,d\tau=(-\mathbf{U}_{n}^{T}\mathbf{K}^{-1}% \mathbf{Q}+V_{p})\int_{0}^{t}C_{p}(\tau)\,d\tau+\mathbf{U}_{n}^{T}\mathbf{K}^{% -1}\mathbf{A}

NTCIR12-MathWiki-16rate: 0

0 t ROI ( τ ) d τ ROI ( t ) superscript subscript 0 t ROI τ d τ ROI t {{\int_{0}^{t}\mathrm{ROI}(\tau)d\tau}\over\mathrm{ROI}(t)}

NTCIR12-MathWiki-16rate: 1

0 y λ J λ d λ 0 J λ d λ , subscript superscript 0 subscript y λ subscript J λ d λ subscript superscript 0 subscript J λ d λ \frac{\int^{\infty}_{0}y_{\lambda}J_{\lambda}d\lambda}{\int^{\infty}_{0}J_{% \lambda}d\lambda},

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Ω 1 = 0 t A ( τ ) d τ subscript normal-Ω 1 superscript subscript 0 t A τ d τ \Omega_{1}=\int_{0}^{t}A(\tau)d\tau

NTCIR12-MathWiki-16rate: 2

g ( s ) = 2 / π 0 ( s t ) 1 / 2 K ν ( s t ) f ( t ) d t , g s 2 π superscript subscript 0 superscript s t 1 2 subscript K ν s t f t d t g(s)=\sqrt{2/\pi}\int_{0}^{\infty}(st)^{1/2}\,K_{\nu}(st)\,f(t)\;dt,

NTCIR12-MathWiki-16rate: 0

m ( φ ) = 0 φ M ( φ ) d φ = a ( 1 - e 2 ) 0 φ ( 1 - e 2 sin 2 φ ) - 3 / 2 d φ . m φ absent superscript subscript 0 φ M φ d φ a 1 superscript e 2 superscript subscript 0 φ superscript 1 superscript e 2 superscript 2 φ 3 2 d φ \begin{aligned}\displaystyle m(\varphi)&\displaystyle=\int_{0}^{\varphi}M(% \varphi)\,d\varphi=a(1-e^{2})\int_{0}^{\varphi}\bigl(1-e^{2}\sin^{2}\varphi% \bigr)^{-3/2}\,d\varphi.\end{aligned}

NTCIR12-MathWiki-16rate: 1

μ n = - ( x - c ) n f ( x ) d x . subscript μ n superscript subscript superscript x c n f x d x \mu_{n}=\int_{-\infty}^{\infty}(x-c)^{n}\,f(x)\,dx.

NTCIR12-MathWiki-16rate: 1

ν r = ( x - μ ) r f ( x ) d x subscript ν r superscript x μ r f x d x \nu_{r}=\int(x-\mu)^{r}f(x)\,dx

NTCIR12-MathWiki-16rate: 1

S ( τ B ) - S ( τ A ) = A B d S = τ A τ B d S d τ d τ = τ A τ B S ( τ + d τ ) - S ( τ ) ( τ + d τ ) - τ d τ S subscript τ B S subscript τ A superscript subscript A B d S superscript subscript subscript τ A subscript τ B d S d τ d τ superscript subscript subscript τ A subscript τ B S τ d τ S τ τ d τ τ d τ S(\tau_{B})-S(\tau_{A})=\int_{A}^{B}dS=\int_{\tau_{A}}^{\tau_{B}}\frac{dS}{d% \tau}d\tau=\int_{\tau_{A}}^{\tau_{B}}\frac{S(\tau+d\tau)-S(\tau)}{(\tau+d\tau)% -\tau}d\tau

NTCIR12-MathWiki-16rate: 1

Σ ( x - x ¯ ) 2 n = Σ [ x - ( Σ x ) / n ] 2 n , normal-Σ superscript x normal-¯ x 2 n normal-Σ superscript delimited-[] x normal-Σ x n 2 n \sqrt{\frac{\Sigma(x-\bar{x})^{2}}{n}}=\sqrt{\frac{\Sigma\left[x-\left(\Sigma x% \right)/n\right]^{2}}{n}}\ ,

NTCIR12-MathWiki-16rate: 3

Ω ( r ) = r ( 1 - F ( x ) ) d x - r F ( x ) d x normal-Ω r superscript subscript r 1 F x d x superscript subscript r F x d x \Omega(r)=\frac{\int_{r}^{\infty}(1-F(x))\,dx}{\int_{-\infty}^{r}F(x)dx}

NTCIR12-MathWiki-16rate: 2

1 κ = 0 κ ν - 1 u ( ν , T ) d ν 0 u ( ν , T ) d ν 1 κ superscript subscript 0 superscript subscript κ ν 1 u ν T d ν superscript subscript 0 u ν T d ν \frac{1}{\kappa}=\frac{\int_{0}^{\infty}\kappa_{\nu}^{-1}u(\nu,T)d\nu}{\int_{0% }^{\infty}u(\nu,T)d\nu}

NTCIR12-MathWiki-16rate: 4

1 κ = 0 ( κ ν , es + κ ν , ff ) - 1 u ( ν , T ) d ν 0 u ( ν , T ) d ν 1 κ superscript subscript 0 superscript subscript κ ν es subscript κ ν ff 1 u ν T d ν superscript subscript 0 u ν T d ν \frac{1}{\kappa}=\frac{\int_{0}^{\infty}(\kappa_{\nu,{\rm es}}+\kappa_{\nu,{% \rm ff}})^{-1}u(\nu,T)d\nu}{\int_{0}^{\infty}u(\nu,T)d\nu}

NTCIR12-MathWiki-16rate: 1

Var ( x ) = - ( x - x ) 2 P n ( x ) d x = L 2 12 ( 1 - 6 n 2 π 2 ) Var x superscript subscript superscript x delimited-⟨⟩ x 2 subscript P n x d x superscript L 2 12 1 6 superscript n 2 superscript π 2 \mathrm{Var}(x)=\int_{-\infty}^{\infty}(x-\langle x\rangle)^{2}P_{n}(x)\,dx=% \frac{L^{2}}{12}\left(1-\frac{6}{n^{2}\pi^{2}}\right)

NTCIR12-MathWiki-16rate: 1

G ( R , R ; N ) R 0 = R R N = R 𝒟 R ( n ) exp [ - 3 2 l 2 0 N d n ( R n n ) 2 - β 0 N d u U e [ R ( n ) ] ] d R d R R 0 = R R N = R 𝒟 R n exp [ - 3 2 l 2 0 N d n ( R n n ) 2 ] G normal-→ R superscript normal-→ R normal-′ N superscript subscript subscript normal-→ R 0 superscript normal-→ R normal-′ subscript normal-→ R N normal-→ R 𝒟 normal-→ R n 3 2 superscript l 2 superscript subscript 0 N d n superscript subscript normal-→ R n n 2 β superscript subscript 0 N d u subscript U e delimited-[] normal-→ R n d superscript normal-→ R normal-′ d normal-→ R superscript subscript subscript normal-→ R 0 superscript normal-→ R normal-′ subscript normal-→ R N normal-→ R 𝒟 subscript normal-→ R n 3 2 superscript l 2 superscript subscript 0 N d n superscript subscript normal-→ R n n 2 G(\vec{R},\vec{R}^{\prime};N)\equiv\frac{\displaystyle\int_{\vec{R}_{0}=\vec{R% }^{\prime}}^{\vec{R}_{N}=\vec{R}}\mathcal{D}\vec{R}(n)\exp\left[-\frac{3}{2l^{% 2}}\displaystyle\int_{0}^{N}dn\left(\frac{\partial\vec{R}_{n}}{\partial n}% \right)^{2}-\beta\displaystyle\int_{0}^{N}duU_{e}[\vec{R}(n)]\right]}{% \displaystyle\int d\vec{R}^{\prime}\displaystyle\int d\vec{R}\displaystyle\int% _{\vec{R}_{0}=\vec{R}^{\prime}}^{\vec{R}_{N}=\vec{R}}\mathcal{D}\vec{R}_{n}% \exp\left[-\frac{3}{2l^{2}}\displaystyle\int_{0}^{N}dn\left(\frac{\partial\vec% {R}_{n}}{\partial n}\right)^{2}\right]}

NTCIR12-MathWiki-16rate: 1

R ( t ) = K 0 t C p ( τ ) d τ + V 0 C p ( t ) R t K superscript subscript 0 t subscript C p τ d τ subscript V 0 subscript C p t R(t)=K\int_{0}^{t}C_{p}(\tau)\,d\tau+V_{0}C_{p}(t)

NTCIR12-MathWiki-16rate: 1

u = 2 ( p t - p s ) ρ u 2 subscript p t subscript p s ρ u=\sqrt{\frac{2(p_{t}-p_{s})}{\rho}}

NTCIR12-MathWiki-16rate: 2

d = - t ( t - r ) 2 f ( r ) d r d superscript subscript t superscript t r 2 f r d r d=\sqrt{\int_{-\infty}^{t}(t-r)^{2}f(r)\,dr}

NTCIR12-MathWiki-16rate: 3

f X Y = y ( x ) = f X ( x ) L X Y = y ( x ) - f X ( x ) L X Y = y ( x ) d x subscript f fragments X normal-∣ Y y x subscript f X x subscript L fragments X normal-∣ Y y x superscript subscript subscript f X x subscript L fragments X normal-∣ Y y x d x f_{X\mid Y=y}(x)={f_{X}(x)L_{X\mid Y=y}(x)\over{\int_{-\infty}^{\infty}f_{X}(x% )L_{X\mid Y=y}(x)\,dx}}

NTCIR12-MathWiki-16rate: 0

I rms subscript I rms I_{\mbox{rms}~{}}

NTCIR12-MathWiki-16rate: 1

- x x ( x 2 - y 2 ) n cos ( y ) d y . superscript subscript x x superscript superscript x 2 superscript y 2 n y d y \int_{-x}^{x}(x^{2}-y^{2})^{n}\cos(y)\,dy.

NTCIR12-MathWiki-16rate: 1

J n ( π 2 ) = - π / 2 π / 2 ( π 2 4 - y 2 ) n cos ( y ) d y = 0 π ( π 2 4 - ( y - π 2 ) 2 ) n cos ( y - π 2 ) d y = 0 π y n ( π - y ) n sin ( y ) d y = n ! b n 0 π f ( x ) sin ( x ) d x . subscript J n π 2 absent superscript subscript π 2 π 2 superscript superscript π 2 4 superscript y 2 n y d y missing-subexpression absent superscript subscript 0 π superscript superscript π 2 4 superscript y π 2 2 n y π 2 d y missing-subexpression absent superscript subscript 0 π superscript y n superscript π y n y d y missing-subexpression absent n superscript b n superscript subscript 0 π f x x d x \begin{aligned}\displaystyle J_{n}\left(\frac{\pi}{2}\right)&\displaystyle=% \int_{-\pi/2}^{\pi/2}\left(\frac{\pi^{2}}{4}-y^{2}\right)^{n}\cos(y)\,dy\\ &\displaystyle=\int_{0}^{\pi}\left(\frac{\pi^{2}}{4}-\left(y-\frac{\pi}{2}% \right)^{2}\right)^{n}\cos\left(y-\frac{\pi}{2}\right)\,dy\\ &\displaystyle=\int_{0}^{\pi}y^{n}(\pi-y)^{n}\sin(y)\,dy\\ &\displaystyle=\frac{n!}{b^{n}}\int_{0}^{\pi}f(x)\sin(x)\,dx.\end{aligned}

NTCIR12-MathWiki-16rate: 1

d k = b k - 1 b k ( x - y k ) 2 f ( x ) d x subscript d k superscript subscript subscript b k 1 subscript b k superscript x subscript y k 2 f x d x d_{k}=\int_{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx

NTCIR12-MathWiki-16rate: 0

t ρ I ( t ) = - 1 2 m , n 0 d τ ( C m n ( τ ) [ S m , I ( t ) , S n , I ( t - τ ) ρ I ( t ) ] - C m n ( τ ) [ S m , I ( t ) , ρ I ( t ) S n , I ( t - τ ) ] ) t subscript ρ I t 1 superscript Planck-constant-over-2-pi 2 subscript m n superscript subscript 0 d τ subscript C m n τ subscript S m I t subscript S n I t τ subscript ρ I t superscript subscript C m n normal-∗ τ subscript S m I t subscript ρ I t subscript S n I t τ \frac{\partial}{\partial t}\rho_{I}(t)=-\frac{1}{\hbar^{2}}\sum_{m,n}\int_{0}^% {\infty}d\tau\biggl(C_{mn}(\tau)\Bigl[S_{m,I}(t),S_{n,I}(t-\tau)\rho_{I}(t)% \Bigr]-C_{mn}^{\ast}(\tau)\Bigl[S_{m,I}(t),\rho_{I}(t)S_{n,I}(t-\tau)\Bigr]\biggr)

NTCIR12-MathWiki-16rate: 1

σ 2 = 0 ( t - t ¯ ) 2 E ( t ) d t superscript σ 2 superscript subscript 0 normal-⋅ superscript t normal-¯ t 2 E t d t \sigma^{2}=\int_{0}^{\infty}(t-\bar{t})^{2}\cdot E(t)\,dt

NTCIR12-MathWiki-16rate: 2

E ( t ) = C ( t ) 0 C ( t ) d t E t C t superscript subscript 0 C t d t E(t)=\frac{C(t)}{\int_{0}^{\infty}C(t)\ dt}

NTCIR12-MathWiki-16rate: 1

RMSD = t = 1 n ( y ^ t - y ) 2 n . RMSD superscript subscript t 1 n superscript subscript normal-^ y t y 2 n \operatorname{RMSD}=\sqrt{\frac{\sum_{t=1}^{n}(\hat{y}_{t}-y)^{2}}{n}}.

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- w ( τ ) d τ = 1. superscript subscript w τ d τ 1. \int_{-\infty}^{\infty}w(\tau)\,d\tau=1.

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I 1 = τ 0 H ( μ - τ x ) d x - τ 0 H ( μ - τ x ) e x + 1 d x subscript I 1 τ superscript subscript 0 H μ τ x normal-d x τ superscript subscript 0 H μ τ x superscript e x 1 normal-d x I_{1}=\tau\int_{0}^{\infty}H(\mu-\tau x)\,\mathrm{d}x-\tau\int_{0}^{\infty}% \frac{H(\mu-\tau x)}{e^{x}+1}\,\mathrm{d}x\,

NTCIR12-MathWiki-16rate: 2

D R = - T ( T - r ) 2 f ( r ) d r D R superscript subscript T superscript T r 2 f r d r DR=\sqrt{\int_{-\infty}^{T}(T-r)^{2}f(r)\,dr}

NTCIR12-MathWiki-16rate: 0

τ 0 d t e - ( t / τ K ) β = τ K β Γ ( 1 β ) delimited-⟨⟩ τ superscript subscript 0 d t superscript e superscript t subscript τ K β subscript τ K β normal-Γ 1 β \langle\tau\rangle\equiv\int_{0}^{\infty}dt\,e^{-\left({t/\tau_{K}}\right)^{% \beta}}={\tau_{K}\over\beta}\Gamma({1\over\beta})

NTCIR12-MathWiki-16rate: 1

𝐇 α ( x ) = 2 ( x 2 ) α π Γ ( α + 1 2 ) 0 π 2 sin ( x cos τ ) sin 2 α ( τ ) d τ . subscript 𝐇 α x 2 superscript x 2 α π normal-Γ α 1 2 superscript subscript 0 π 2 x τ superscript 2 α τ d τ \mathbf{H}_{\alpha}(x)=\frac{2\left(\frac{x}{2}\right)^{\alpha}}{\sqrt{\pi}% \Gamma\left(\alpha+\frac{1}{2}\right)}\int_{0}^{\frac{\pi}{2}}\sin(x\cos\tau)% \sin^{2\alpha}(\tau)d\tau.

NTCIR12-MathWiki-16rate: 1

A = 2 π 0 π sin ( t ) ( cos ( t ) ) 2 + ( sin ( t ) ) 2 d t = 2 π 0 π sin ( t ) d t = 4 π . A absent 2 π superscript subscript 0 π t superscript t 2 superscript t 2 d t missing-subexpression absent 2 π superscript subscript 0 π t d t missing-subexpression absent 4 π \begin{aligned}\displaystyle A&\displaystyle{}=2\pi\int_{0}^{\pi}\sin(t)\sqrt{% \left(\cos(t)\right)^{2}+\left(\sin(t)\right)^{2}}\,dt\\ &\displaystyle{}=2\pi\int_{0}^{\pi}\sin(t)\,dt\\ &\displaystyle{}=4\pi.\end{aligned}

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T = e - i = 1 N σ i 0 n i ( z ) d z = 10 - i = 1 N ε i 0 c i ( z ) d z , T superscript e superscript subscript i 1 N subscript σ i superscript subscript 0 normal-ℓ subscript n i z normal-d z superscript 10 superscript subscript i 1 N subscript ε i superscript subscript 0 normal-ℓ subscript c i z normal-d z T=e^{-\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\mathrm{d}z}=10^{-\sum_{i% =1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\mathrm{d}z},

NTCIR12-MathWiki-16rate: 2

Δ t 2 = [ 0 Δ τ e 0 τ ¯ a ( τ ) d τ d τ ¯ ] [ 0 Δ τ e - 0 τ ¯ a ( τ ) d τ d τ ¯ ] , normal-Δ superscript t 2 delimited-[] subscript superscript normal-Δ τ 0 superscript e subscript superscript normal-¯ τ 0 a superscript τ normal-′ d superscript τ normal-′ d normal-¯ τ delimited-[] subscript superscript normal-Δ τ 0 superscript e subscript superscript normal-¯ τ 0 a superscript τ normal-′ d superscript τ normal-′ d normal-¯ τ \Delta t^{2}=\left[\int^{\Delta\tau}_{0}e^{\int^{\bar{\tau}}_{0}a(\tau^{\prime% })d\tau^{\prime}}\,d\bar{\tau}\right]\,\left[\int^{\Delta\tau}_{0}e^{-\int^{% \bar{\tau}}_{0}a(\tau^{\prime})d\tau^{\prime}}\,d\bar{\tau}\right],

NTCIR12-MathWiki-16rate: 1

( 3 ) 0 δ e - x t ϕ ( t ) d t = 0 δ e - x t t λ g ( t ) d t = n = 0 N g ( n ) ( 0 ) n ! 0 δ t λ + n e - x t d t + 1 ( N + 1 ) ! 0 δ g ( N + 1 ) ( t * ) t λ + N + 1 e - x t d t . 3 superscript subscript 0 δ superscript e x t ϕ t normal-d t absent superscript subscript 0 δ superscript e x t superscript t λ g t normal-d t missing-subexpression absent superscript subscript n 0 N superscript g n 0 n superscript subscript 0 δ superscript t λ n superscript e x t normal-d t 1 N 1 superscript subscript 0 δ superscript g N 1 superscript t superscript t λ N 1 superscript e x t normal-d t \begin{aligned}\displaystyle(3)\quad\int_{0}^{\delta}e^{-xt}\phi(t)\,\mathrm{d% }t&\displaystyle=\int_{0}^{\delta}e^{-xt}t^{\lambda}g(t)\,\mathrm{d}t\\ &\displaystyle=\sum_{n=0}^{N}\frac{g^{(n)}(0)}{n!}\int_{0}^{\delta}t^{\lambda+% n}e^{-xt}\,\mathrm{d}t+\frac{1}{(N+1)!}\int_{0}^{\delta}g^{(N+1)}(t^{*})\,t^{% \lambda+N+1}e^{-xt}\,\mathrm{d}t.\end{aligned}

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s = i = 1 n ( x i - x ¯ ) 2 n - 1 s superscript subscript i 1 n superscript subscript x i normal-¯ x 2 n 1 s=\sqrt{\frac{\sum_{i=1}^{n}{\left(x_{i}-\bar{x}\right)}^{2}}{n-1}}

NTCIR12-MathWiki-16rate: 2

v 3 ( t ) = K 1 t 0 t i 1 ( τ ) d τ + K 2 t 0 t i 2 ( τ ) d τ , subscript v 3 t subscript K 1 superscript subscript subscript t 0 t subscript i 1 τ d τ subscript K 2 superscript subscript subscript t 0 t subscript i 2 τ d τ v_{3}(t)=K_{1}\int_{t_{0}}^{t}i_{1}(\tau)d\tau+K_{2}\int_{t_{0}}^{t}i_{2}(\tau% )d\tau,

NTCIR12-MathWiki-17rate: 0

1 - 1 2 - 1 4 + 1 8 - 1 16 + = 1 3 . 1 1 2 1 4 1 8 1 16 normal-⋯ 1 3 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1}{3}.

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c = - 1 6 1 2 ! = - 1 12 . c normal-⋅ 1 6 1 2 1 12 c=-\frac{1}{6}\cdot\frac{1}{2!}=-\frac{1}{12}.

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1 2 a 0 - 1 4 Δ a 0 + 1 8 Δ 2 a 0 - = 1 2 - 1 4 . 1 2 subscript a 0 1 4 normal-Δ subscript a 0 1 8 superscript normal-Δ 2 subscript a 0 normal-⋯ 1 2 1 4 \frac{1}{2}a_{0}-\frac{1}{4}\Delta a_{0}+\frac{1}{8}\Delta^{2}a_{0}-\cdots=% \frac{1}{2}-\frac{1}{4}.

NTCIR12-MathWiki-17rate: 1

a 0 2 - Δ a 0 4 + Δ 2 a 0 8 - Δ 3 a 0 16 + = 1 2 - 1 4 + 1 8 - 1 16 + . subscript a 0 2 normal-Δ subscript a 0 4 superscript normal-Δ 2 subscript a 0 8 superscript normal-Δ 3 subscript a 0 16 normal-⋯ 1 2 1 4 1 8 1 16 normal-⋯ \frac{a_{0}}{2}-\frac{\Delta a_{0}}{4}+\frac{\Delta^{2}a_{0}}{8}-\frac{\Delta^% {3}a_{0}}{16}+\cdots=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots.

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η ( s ) = n = 1 ( - 1 ) n - 1 n s = 1 1 s - 1 2 s + 1 3 s - 1 4 s + η s superscript subscript n 1 superscript 1 n 1 superscript n s 1 superscript 1 s 1 superscript 2 s 1 superscript 3 s 1 superscript 4 s normal-⋯ \eta(s)=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^% {s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots

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g = ( 1 , 0 , 1 , - 1 2 , 1 3 , - 1 4 , 1 5 , - 1 6 , ) g 1 0 1 1 2 1 3 1 4 1 5 1 6 normal-⋯ g=\left(1,0,1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},% \cdots\right)

NTCIR12-MathWiki-17rate: 0

π = 1 2 6 n = 0 ( - 1 ) n 2 10 n ( - 2 5 4 n + 1 - 1 4 n + 3 + 2 8 10 n + 1 - 2 6 10 n + 3 - 2 2 10 n + 5 - 2 2 10 n + 7 + 1 10 n + 9 ) π 1 superscript 2 6 superscript subscript n 0 superscript 1 n superscript 2 10 n superscript 2 5 4 n 1 1 4 n 3 superscript 2 8 10 n 1 superscript 2 6 10 n 3 superscript 2 2 10 n 5 superscript 2 2 10 n 7 1 10 n 9 \pi=\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{10n}}\left(-\frac{2^{% 5}}{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}}{% 10n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)\!

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π = 1 2 6 n = 0 ( - 1 ) n 2 10 n ( - 2 5 4 n + 1 - 1 4 n + 3 + 2 8 10 n + 1 - 2 6 10 n + 3 - 2 2 10 n + 5 - 2 2 10 n + 7 + 1 10 n + 9 ) π 1 superscript 2 6 superscript subscript n 0 superscript 1 n superscript 2 10 n superscript 2 5 4 n 1 1 4 n 3 superscript 2 8 10 n 1 superscript 2 6 10 n 3 superscript 2 2 10 n 5 superscript 2 2 10 n 7 1 10 n 9 \pi=\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{{(-1)}^{n}}{2^{10n}}\left(-\frac{2% ^{5}}{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}% }{10n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)\!

NTCIR12-MathWiki-17rate: 1

( - 3 2 ) + ( - 1 2 ) + 1 2 = 3 ( - 3 2 + 1 2 ) 2 = - 3 2 . 3 2 1 2 1 2 3 3 2 1 2 2 3 2 \left(-\frac{3}{2}\right)+\left(-\frac{1}{2}\right)+\frac{1}{2}=\frac{3\left(-% \frac{3}{2}+\frac{1}{2}\right)}{2}=-\frac{3}{2}.

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M O = 1 - 3 5 = 2 5 subscript M O 1 3 5 2 5 M_{O}=1-\tfrac{3}{5}=\tfrac{2}{5}

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B 1 = - 1 2 subscript B 1 1 2 \scriptstyle B_{1}=-{1\over 2}

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V = 2 ( 1 r 12 - 1 r 1 a - 1 r 1 b - 1 r 2 a - 1 r 2 b + 1 r a b ) V 2 1 subscript r 12 1 subscript r 1 a 1 subscript r 1 b 1 subscript r 2 a 1 subscript r 2 b 1 subscript r a b V=2(\frac{1}{r_{12}}-\frac{1}{r_{1a}}-\frac{1}{r_{1b}}-\frac{1}{r_{2a}}-\frac{% 1}{r_{2b}}+\frac{1}{r_{ab}})

NTCIR12-MathWiki-17rate: 2

( 1 - x 2 ) 1 / 3 = 1 - x 2 3 - x 4 9 - 5 x 6 81 superscript 1 superscript x 2 1 3 1 superscript x 2 3 superscript x 4 9 5 superscript x 6 81 normal-⋯ (1-x^{2})^{1/3}=1-\frac{x^{2}}{3}-\frac{x^{4}}{9}-\frac{5x^{6}}{81}\cdots

NTCIR12-MathWiki-17rate: 1

C = ( - 1 ) i s i - 1 = 1 1 - 1 2 + 1 6 - 1 42 + 1 1806 - 0.64341054629. C superscript 1 i subscript s i 1 1 1 1 2 1 6 1 42 1 1806 normal-⋯ 0.64341054629. C=\sum\frac{(-1)^{i}}{s_{i}-1}=\frac{1}{1}-\frac{1}{2}+\frac{1}{6}-\frac{1}{42% }+\frac{1}{1806}-\cdots\approx 0.64341054629.

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1 - 1 2 + 1 3 - 1 4 + 1 5 - = n = 1 ( - 1 ) n + 1 n 1 1 2 1 3 1 4 1 5 normal-⋯ superscript subscript n 1 superscript 1 n 1 n 1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-\cdots=\sum\limits_{n=1}^{\infty% }{(-1)^{n+1}\over n}

NTCIR12-MathWiki-17rate: 1

1 1 - 1 2 + 1 4 - 1 8 + 1 16 - 1 32 + = 2 3 . 1 1 1 2 1 4 1 8 1 16 1 32 normal-⋯ 2 3 {1\over 1}-{1\over 2}+{1\over 4}-{1\over 8}+{1\over 16}-{1\over 32}+\cdots={2% \over 3}.

NTCIR12-MathWiki-17rate: 1

1 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + . normal-→ 1 1 1 2 1 3 1 4 1 5 1 6 normal-⋯ {1\over 1}+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+{1\over 6}+\cdots% \rightarrow\infty.

NTCIR12-MathWiki-17rate: 1

ψ ( x ) = ln ( x ) - 1 2 x - 1 12 x 2 + 1 120 x 4 - 1 252 x 6 + 1 240 x 8 - 5 660 x 10 + 691 32760 x 12 - 1 12 x 14 + O ( 1 x 16 ) ψ x x 1 2 x 1 12 superscript x 2 1 120 superscript x 4 1 252 superscript x 6 1 240 superscript x 8 5 660 superscript x 10 691 32760 superscript x 12 1 12 superscript x 14 O 1 superscript x 16 \psi(x)=\ln(x)-\frac{1}{2x}-\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac{1}{252x% ^{6}}+\frac{1}{240x^{8}}-\frac{5}{660x^{10}}+\frac{691}{32760x^{12}}-\frac{1}{% 12x^{14}}+O\left(\frac{1}{x^{16}}\right)

NTCIR12-MathWiki-17rate: 0

η ( s ) = n = 1 ( - 1 ) n - 1 n s = 1 1 s - 1 2 s + 1 3 s - 1 4 s + η s superscript subscript n 1 superscript 1 n 1 superscript n s 1 superscript 1 s 1 superscript 2 s 1 superscript 3 s 1 superscript 4 s normal-⋯ \eta(s)=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^% {s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots

NTCIR12-MathWiki-17rate: 1

1 + 1 2 + 1 3 + 1 4 + 1 5 + = n = 1 1 n . 1 1 2 1 3 1 4 1 5 normal-⋯ superscript subscript n 1 1 n 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots=\sum_{n=1}^{\infty}% \frac{1}{n}.

NTCIR12-MathWiki-17rate: 1

1 2 - 1 4 = 1 4 1 2 1 4 1 4 \frac{1}{2}-\frac{1}{4}=\frac{1}{4}

NTCIR12-MathWiki-17rate: 1

γ = k = 2 ( - 1 ) k log 2 k k = 1 2 - 1 3 + 2 ( 1 4 - 1 5 + 1 6 - 1 7 ) + 3 ( 1 8 - 1 9 + 1 10 - 1 11 + - 1 15 ) + γ superscript subscript k 2 superscript 1 k subscript 2 k k 1 2 1 3 2 1 4 1 5 1 6 1 7 3 1 8 1 9 1 10 1 11 normal-… 1 15 normal-… {\gamma=\sum_{k=2}^{\infty}(-1)^{k}\frac{\left\lfloor\log_{2}k\right\rfloor}{k% }=\frac{1}{2}-\frac{1}{3}+2\left(\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{% 7}\right)+3\left(\frac{1}{8}-\frac{1}{9}+\frac{1}{10}-\frac{1}{11}+\dots-\frac% {1}{15}\right)+\dots}

NTCIR12-MathWiki-17rate: 1

1 2 ( 0 - 6 4 ) = - 3 4 1 2 0 6 4 3 4 \tfrac{1}{2}(0-\tfrac{6}{4})=-\tfrac{3}{4}

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B 1 = - 1 2 subscript B 1 1 2 B_{1}=-\frac{1}{2}

NTCIR12-MathWiki-17rate: 1

1 2 + 1 4 + 1 8 + 1 16 + = 1 1 2 1 4 1 8 1 16 normal-⋯ 1 \frac{1}{2}\,+\,\frac{1}{4}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,\cdots\;=\;1

NTCIR12-MathWiki-17rate: 1

2 3 - 1 2 = 4 6 - 3 6 = 1 6 2 3 1 2 4 6 3 6 1 6 \tfrac{2}{3}-\tfrac{1}{2}=\tfrac{4}{6}-\tfrac{3}{6}=\tfrac{1}{6}

NTCIR12-MathWiki-17rate: 0

π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + = 4 - 1 + 1 6 - 1 34 + 16 3145 - 4 4551 + 1 6601 - 1 38341 + - fragments π continued-fraction 4 1 continued-fraction superscript 1 2 3 continued-fraction superscript 2 2 5 continued-fraction superscript 3 2 7 normal-⋱ 4 1 1 6 1 34 16 3145 4 4551 1 6601 1 38341 normal-⋯ \pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}=4-1% +\frac{1}{6}-\frac{1}{34}+\frac{16}{3145}-\frac{4}{4551}+\frac{1}{6601}-\frac{% 1}{38341}+-\cdots

NTCIR12-MathWiki-17rate: 1

1 2 - 1 4 + 1 8 - 1 16 + = 1 / 2 1 - ( - 1 / 2 ) = 1 3 . 1 2 1 4 1 8 1 16 normal-⋯ 1 2 1 1 2 1 3 \frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)}=% \frac{1}{3}.

NTCIR12-MathWiki-17rate: 4

x - 1 - 1 2 - 1 4 - 1 5 - 1 6 - 1 9 - = 1 x 1 1 2 1 4 1 5 1 6 1 9 normal-⋯ 1 x-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}-\frac{1}{9}-\cdots=1

NTCIR12-MathWiki-17rate: 0

H 1 4 = 4 - π 2 - 3 ln 2 subscript H 1 4 4 π 2 3 2 H_{\frac{1}{4}}=4-\tfrac{\pi}{2}-3\ln{2}

NTCIR12-MathWiki-17rate: 1

1 - 1 2 + 1 3 - 1 4 + 1 5 - = ln 2. 1 1 2 1 3 1 4 1 5 normal-⋯ 2. 1\,-\,\frac{1}{2}\,+\,\frac{1}{3}\,-\,\frac{1}{4}\,+\,\frac{1}{5}\,-\,\cdots\;% =\;\ln 2.

NTCIR12-MathWiki-17rate: 1

1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 + 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 normal-⋯ \displaystyle 1\;\;+\;\;\frac{1}{2}\;\;+\;\;\frac{1}{3}\,+\,\frac{1}{4}\;\;+\;% \;\frac{1}{5}\,+\,\frac{1}{6}\,+\,\frac{1}{7}\,+\,\frac{1}{8}\;\;+\;\;\frac{1}% {9}\,+\,\cdots

NTCIR12-MathWiki-17rate: 0

H = [ 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 1 3 1 4 1 5 1 6 1 7 1 4 1 5 1 6 1 7 1 8 1 5 1 6 1 7 1 8 1 9 ] . H 1 1 2 1 3 1 4 1 5 1 2 1 3 1 4 1 5 1 6 1 3 1 4 1 5 1 6 1 7 1 4 1 5 1 6 1 7 1 8 1 5 1 6 1 7 1 8 1 9 H=\begin{bmatrix}1&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\frac{1}{5}\\ \frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\frac{1}{5}&\frac{1}{6}\\ \frac{1}{3}&\frac{1}{4}&\frac{1}{5}&\frac{1}{6}&\frac{1}{7}\\ \frac{1}{4}&\frac{1}{5}&\frac{1}{6}&\frac{1}{7}&\frac{1}{8}\\ \frac{1}{5}&\frac{1}{6}&\frac{1}{7}&\frac{1}{8}&\frac{1}{9}\end{bmatrix}.

NTCIR12-MathWiki-17rate: 1

π 4 = 1 - 1 3 + 1 5 - 1 7 + π 4 1 1 3 1 5 1 7 normal-⋯ \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots

NTCIR12-MathWiki-17rate: 0

π 4 = 3 4 + 1 3 3 - 3 - 1 5 3 - 5 + 1 7 3 - 7 - π 4 3 4 1 superscript 3 3 3 1 superscript 5 3 5 1 superscript 7 3 7 normal-⋯ \frac{\pi}{4}=\frac{3}{4}+\frac{1}{3^{3}-3}-\frac{1}{5^{3}-5}+\frac{1}{7^{3}-7% }-\cdots

NTCIR12-MathWiki-17rate: 0

β = - 1 2 β 1 2 \beta=-\frac{1}{2}

NTCIR12-MathWiki-17rate: 0

= μ 0 r 2 N 2 π l ( 1 - 8 w 3 π + w 2 2 - w 4 4 + 5 w 6 16 - 35 w 8 64 + ) absent subscript μ 0 superscript r 2 superscript N 2 π l 1 8 w 3 π superscript w 2 2 superscript w 4 4 5 superscript w 6 16 35 superscript w 8 64 normal-… =\frac{\mu_{0}r^{2}N^{2}\pi}{l}\left(1-\frac{8w}{3\pi}+\frac{w^{2}}{2}-\frac{w% ^{4}}{4}+\frac{5w^{6}}{16}-\frac{35w^{8}}{64}+...\right)

NTCIR12-MathWiki-17rate: 0

( [ - 1 , 1 ] + 1 2 ) 2 - 1 4 = [ - 1 2 , 3 2 ] 2 - 1 4 = [ 0 , 9 4 ] - 1 4 = [ - 1 4 , 2 ] superscript 1 1 1 2 2 1 4 superscript 1 2 3 2 2 1 4 0 9 4 1 4 1 4 2 \left([-1,1]+\frac{1}{2}\right)^{2}-\frac{1}{4}=\left[-\frac{1}{2},\frac{3}{2}% \right]^{2}-\frac{1}{4}=\left[0,\frac{9}{4}\right]-\frac{1}{4}=\left[-\frac{1}% {4},2\right]

NTCIR12-MathWiki-17rate: 0

1 r min - 1 p = 1 p - 1 r max 1 subscript r 1 p 1 p 1 subscript r \frac{1}{r_{\min}}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_{\max}}

NTCIR12-MathWiki-17rate: 1

π 4 = 1 - 1 3 + 1 5 - 1 7 + π 4 1 1 3 1 5 1 7 normal-… \frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\ldots

NTCIR12-MathWiki-17rate: 0

π 4 = 3 4 + 1 3 3 - 3 - 1 5 3 - 5 + 1 7 3 - 7 - π 4 3 4 1 superscript 3 3 3 1 superscript 5 3 5 1 superscript 7 3 7 normal-⋯ \frac{\pi}{4}=\frac{3}{4}+\frac{1}{3^{3}-3}-\frac{1}{5^{3}-5}+\frac{1}{7^{3}-7% }-\cdots

NTCIR12-MathWiki-17rate: 0

W ( e 1 + z ) = 1 + z 2 + z 2 16 - z 3 192 - z 4 3072 + 13 z 5 61440 - 47 z 6 1474560 - 73 z 7 41287680 + 2447 z 8 1321205760 + O ( z 9 ) . W superscript e 1 z 1 z 2 superscript z 2 16 superscript z 3 192 superscript z 4 3072 13 superscript z 5 61440 47 superscript z 6 1474560 73 superscript z 7 41287680 2447 superscript z 8 1321205760 O superscript z 9 W(e^{1+z})=1+\frac{z}{2}+\frac{z^{2}}{16}-\frac{z^{3}}{192}-\frac{z^{4}}{3072}% +\frac{13z^{5}}{61440}-\frac{47z^{6}}{1474560}-\frac{73z^{7}}{41287680}+\frac{% 2447z^{8}}{1321205760}+O(z^{9}).

NTCIR12-MathWiki-17rate: 2

1 - 1 3 + 1 5 - 1 7 + 1 9 - = π 4 . 1 1 3 1 5 1 7 1 9 normal-⋯ π 4 1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;% =\;\frac{\pi}{4}.\!

NTCIR12-MathWiki-17rate: 0

μ 3 , 1 = μ 3 , 1 - r = 14 3 - 5 = - 1 3 subscript μ 3 1 subscript μ 3 1 r 14 3 5 1 3 \mu_{3,1}=\mu_{3,1}-r=\frac{14}{3}-5=\frac{-1}{3}

NTCIR12-MathWiki-17rate: 0

1 2 6 n = 0 ( - 1 ) n 2 10 n ( - 2 5 4 n + 1 - 1 4 n + 3 + 2 8 10 n + 1 - 2 6 10 n + 3 - 2 2 10 n + 5 - 2 2 10 n + 7 + 1 10 n + 9 ) = π 1 superscript 2 6 superscript subscript n 0 superscript 1 n superscript 2 10 n superscript 2 5 4 n 1 1 4 n 3 superscript 2 8 10 n 1 superscript 2 6 10 n 3 superscript 2 2 10 n 5 superscript 2 2 10 n 7 1 10 n 9 π \frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{{(-1)}^{n}}{2^{10n}}\left(-\frac{2^{5}% }{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}}{10% n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)=\pi\!

NTCIR12-MathWiki-17rate: 1

n = 0 ( ( - 1 ) n 2 n + 1 ) 1 = 1 1 - 1 3 + 1 5 - 1 7 + 1 9 - = arctan 1 = π 4 superscript subscript n 0 superscript superscript 1 n 2 n 1 1 1 1 1 3 1 5 1 7 1 9 normal-⋯ 1 π 4 \sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{1}=\frac{1}{1}-\frac{% 1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots=\arctan{1}=\frac{\pi}{4}\!

NTCIR12-MathWiki-17rate: 1

π = < m t p l > 1 + 1 2 + 1 3 + 1 4 - 1 5 + 1 6 + 1 7 + 1 8 + 1 9 - 1 10 + 1 11 + 1 12 - 1 13 + π expectation m t p l 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 normal-⋯ \pi=<mtpl>{{1}}+\frac{{1}}{{2}}+\frac{{1}}{{3}}+\frac{{1}}{{4}}-\frac{{1}}{{5}% }+\frac{{1}}{{6}}+\frac{{1}}{{7}}+\frac{{1}}{{8}}+\frac{{1}}{{9}}-\frac{{1}}{{% 10}}+\frac{{1}}{{11}}+\frac{{1}}{{12}}-\frac{{1}}{{13}}+\cdots\!

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n = 1 ( 4 n 2 4 n 2 - 1 ) = 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8 9 superscript subscript product n 1 4 superscript n 2 4 superscript n 2 1 normal-⋅ 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8 9 normal-⋯ \prod_{n=1}^{\infty}\left(\frac{4n^{2}}{4n^{2}-1}\right)=\frac{2}{1}\cdot\frac% {2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot% \frac{8}{7}\cdot\frac{8}{9}\cdots

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n = 1 1 n ( 2 n n ) = 1 - 1 2 + 1 4 - 1 5 + 1 7 - 1 8 + superscript subscript n 1 1 n binomial 2 n n 1 1 2 1 4 1 5 1 7 1 8 normal-⋯ \sum_{n=1}^{\infty}\frac{1}{n{2n\choose n}}=1-\frac{1}{2}+\frac{1}{4}-\frac{1}% {5}+\frac{1}{7}-\frac{1}{8}+\cdots

NTCIR12-MathWiki-17rate: 1

ln 2 = 1 1 - 1 2 + 1 3 - 1 4 + 1 5 - . 2 1 1 1 2 1 3 1 4 1 5 normal-⋯ \ln 2=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots.

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x 1 = - 3 4 subscript x 1 3 4 x_{1}=-\tfrac{3}{4}

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α 1 = 1 2 - m subscript α 1 1 2 m \alpha_{1}=\frac{1}{2}-m

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π = 4 1 - 4 3 + 4 5 - 4 7 + 4 9 - 4 11 + 4 13 - π 4 1 4 3 4 5 4 7 4 9 4 11 4 13 normal-⋯ \pi=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+% \frac{4}{13}-\cdots

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= 1 - 1 6 = 5 6 absent 1 1 6 5 6 =1-\tfrac{1}{6}=\tfrac{5}{6}

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1 4 0 1 x 8 ( 1 - x ) 8 1 + x 2 d x = π - 47 171 15 015 1 4 superscript subscript 0 1 superscript x 8 superscript 1 x 8 1 superscript x 2 d x π 47 171 15 015 \frac{1}{4}\int_{0}^{1}\frac{x^{8}(1-x)^{8}}{1+x^{2}}\,dx=\pi-\frac{47\,171}{1% 5\,015}

NTCIR12-MathWiki-17rate: 1

H 2 n - H n log 2 - 1 4 n + 1 16 n 2 - 1 128 n 4 + 1 256 n 6 - 17 4096 n 8 + , similar-to subscript H 2 n subscript H n 2 1 4 n 1 16 superscript n 2 1 128 superscript n 4 1 256 superscript n 6 17 4096 superscript n 8 normal-⋯ H_{2n}-H_{n}\sim\log 2-\frac{1}{4n}+\frac{1}{16n^{2}}-\frac{1}{128n^{4}}+\frac% {1}{256n^{6}}-\frac{17}{4096n^{8}}+\cdots,

NTCIR12-MathWiki-17rate: 3

1 - 1 2 - 1 4 + 1 3 - 1 6 - 1 8 + 1 5 - 1 10 - 1 12 + 1 1 2 1 4 1 3 1 6 1 8 1 5 1 10 1 12 normal-⋯ 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-% \frac{1}{10}-\frac{1}{12}+\cdots

NTCIR12-MathWiki-17rate: 1

1 + 1 2 + 1 3 + 1 4 + 1 5 + = n = 1 1 n . 1 1 2 1 3 1 4 1 5 normal-⋯ superscript subscript n 1 1 n 1+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+\cdots=\sum_{n=1}^{\infty}{1% \over n}.

NTCIR12-MathWiki-17rate: 1

n = 1 ( - 1 ) n + 1 n = 1 - 1 2 + 1 3 - 1 4 + 1 5 - superscript subscript n 1 superscript 1 n 1 n 1 1 2 1 3 1 4 1 5 normal-⋯ \sum\limits_{n=1}^{\infty}{(-1)^{n+1}\over n}=1-{1\over 2}+{1\over 3}-{1\over 4% }+{1\over 5}-\cdots

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1 - 1 3 + 1 5 - 1 7 + 1 9 - = π 4 , 1 1 3 1 5 1 7 1 9 normal-⋯ π 4 1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;% =\;\frac{\pi}{4},\!

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Li 2 ( 1 3 ) - 1 6 Li 2 ( 1 9 ) = π 2 18 - ln 2 3 6 subscript Li 2 1 3 1 6 subscript Li 2 1 9 superscript π 2 18 superscript 2 3 6 \operatorname{Li}_{2}\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_{2}% \left(\frac{1}{9}\right)=\frac{{\pi}^{2}}{18}-\frac{\ln^{2}3}{6}

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1 = 1 2 + 1 3 + 1 7 + 1 43 + 1 1807 + . 1 1 2 1 3 1 7 1 43 1 1807 normal-⋯ 1=\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\frac{1}{1807}+\cdots.

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L 6 λ = - 2 1 + λ + 60 2 + λ - 420 3 + λ + 1120 4 + λ - 1260 5 + λ + 504 6 + λ . subscript L 6 λ 2 1 λ 60 2 λ 420 3 λ 1120 4 λ 1260 5 λ 504 6 λ L_{6}\lambda=-\frac{2}{1+\lambda}+\frac{60}{2+\lambda}-\frac{420}{3+\lambda}+% \frac{1120}{4+\lambda}-\frac{1260}{5+\lambda}+\frac{504}{6+\lambda}.

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i = 1 - ( 1 - 1 2 ) α = 1 - α 2 i 1 1 1 2 α 1 α 2 i=1-(1-\frac{1}{2})\alpha=1-\frac{\alpha}{2}

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1 - 1 3 + 1 5 - 1 7 + = π 4 = 0.7853981 1 1 3 1 5 1 7 normal-⋯ π 4 0.7853981 normal-… \scriptstyle 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{4}=0.7853% 981\ldots

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Γ ( r + k ) k ! Γ ( r ) p r ( 1 - p ) k normal-Γ r k k normal-Γ r superscript p r superscript 1 p k \frac{\Gamma(r+k)}{k!\,\Gamma(r)}\,p^{r}\,(1-p)^{k}\,

NTCIR12-MathWiki-18rate: 3

f ( p ) = ( n + 1 ) ! k ! ( n - k ) ! p k ( 1 - p ) n - k for 0 p 1 f p n 1 k n k superscript p k superscript 1 p n k for 0 p 1 f(p)=\frac{(n+1)!}{k!(n-k)!}p^{k}(1-p)^{n-k}\,\text{ for }0\leq p\leq 1

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P - m = ( - 1 ) m ( - m ) ! ( + m ) ! P m superscript subscript P normal-ℓ m superscript 1 m normal-ℓ m normal-ℓ m superscript subscript P normal-ℓ m P_{\ell}^{-m}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^{m}

NTCIR12-MathWiki-18rate: 1

= n ! j 1 ! j 2 ! j n - k + 1 ! ( x 1 1 ! ) j 1 ( x 2 2 ! ) j 2 ( x n - k + 1 ( n - k + 1 ) ! ) j n - k + 1 , absent n subscript j 1 subscript j 2 normal-⋯ subscript j n k 1 superscript subscript x 1 1 subscript j 1 superscript subscript x 2 2 subscript j 2 normal-⋯ superscript subscript x n k 1 n k 1 subscript j n k 1 =\sum{n!\over j_{1}!j_{2}!\cdots j_{n-k+1}!}\left({x_{1}\over 1!}\right)^{j_{1% }}\left({x_{2}\over 2!}\right)^{j_{2}}\cdots\left({x_{n-k+1}\over(n-k+1)!}% \right)^{j_{n-k+1}},

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θ n ( x ) = n ! ( - 2 ) n L n - 2 n - 1 ( 2 x ) subscript θ n x n superscript 2 n superscript subscript L n 2 n 1 2 x \theta_{n}(x)=\frac{n!}{(-2)^{n}}\,L_{n}^{-2n-1}(2x)

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( n i ) = n ! i ! ( n - i ) ! binomial n i n i n i {n\choose i}=\frac{n!}{i!(n-i)!}

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( n i ) = n ! i ! ( n - i ) ! binomial n i n i n i {n\choose i}=\frac{n!}{i!(n-i)!}

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Var [ X ] = n p ( 1 - p ) . Var X n p 1 p \operatorname{Var}[X]=np(1-p).

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( n k ) = n ! k ! ( n - k ) ! binomial n k n k n k {n\choose k}=\frac{n!}{k!(n-k)!}

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( n k ) = n ! k ! ( n - k ) ! binomial n k n k n k {n\choose k}=\frac{n!}{k!\,(n-k)!}

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α ! = Π i = 1 N ( α i ! ) α superscript subscript normal-Π i 1 N subscript α i \alpha!=\Pi_{i=1}^{N}(\alpha_{i}!)

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w ( n , g ) = ( n + g - 1 ) ! n ! ( g - 1 ) ! . w n g n g 1 n g 1 w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.

NTCIR12-MathWiki-18rate: 1

D f = L ! N f ! ( L - N f ) ! . subscript D f L subscript N f L subscript N f D_{f}=\frac{L!}{N_{f}!(L-N_{f})!}.

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( N N R ) = N ! N R ! ( N - N R ) ! binomial N subscript N R N subscript N R N subscript N R {\left({{N}\atop{N_{R}}}\right)}=\frac{N!}{N_{R}!(N-N_{R})!}

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P N ( N R ) = N ! 2 N N R ! ( N - N R ) ! subscript P N subscript N R N superscript 2 N subscript N R N subscript N R P_{N}(N_{R})=\frac{N!}{2^{N}N_{R}!(N-N_{R})!}

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( n k ) = n ! k ! ( n - k ) ! , binomial n k n k n k {n\choose k}=\frac{n!}{k!\,(n-k)!},

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f ( x ) - P n - 1 ( x ) = f ( n ) ( ξ ) n ! i = 1 n ( x - x i ) f x subscript P n 1 x superscript f n ξ n superscript subscript product i 1 n x subscript x i f(x)-P_{n-1}(x)=\frac{f^{(n)}(\xi)}{n!}\prod_{i=1}^{n}(x-x_{i})

NTCIR12-MathWiki-18rate: 3

f ( r | H = h , T = t ) = ( h + t + 1 ) ! h ! t ! r h ( 1 - r ) t . fragments f fragments normal-( r normal-| H h normal-, T t normal-) h t 1 h t superscript r h superscript fragments normal-( 1 r normal-) t normal-. f(r|H=h,T=t)=\frac{(h+t+1)!}{h!\,\,t!}\;r^{h}\,(1-r)^{t}.\!

NTCIR12-MathWiki-18rate: 1

( n k ) = n ! k ! ( n - k ) ! , binomial n k n k n k {\left({{n}\atop{k}}\right)}=\frac{n!}{k!(n-k)!},

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Ω = ( q + N - 1 ) ! q ! ( N - 1 ) ! normal-Ω q superscript N normal-′ 1 q superscript N normal-′ 1 \Omega={\left(q+N^{\prime}-1\right)!\over q!(N^{\prime}-1)!}

NTCIR12-MathWiki-18rate: 1

E n ( x ) = n ! π 0 x e - t n d t = n ! π p = 0 ( - 1 ) p x n p + 1 ( n p + 1 ) p ! . subscript E n x n π superscript subscript 0 x superscript e superscript t n normal-d t n π superscript subscript p 0 superscript 1 p superscript x n p 1 n p 1 p E_{n}(x)=\frac{n!}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{n}}\,\mathrm{d}t=\frac{n!}{% \sqrt{\pi}}\sum_{p=0}^{\infty}(-1)^{p}\frac{x^{np+1}}{(np+1)p!}\,.

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n ! i = 1 k x i ! n superscript subscript product i 1 k subscript x i \frac{n!}{\prod_{i=1}^{k}x_{i}!}

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w ( n i , g i ) = g i ! n i ! ( g i - n i ) ! . w subscript n i subscript g i subscript g i subscript n i subscript g i subscript n i w(n_{i},g_{i})=\frac{g_{i}!}{n_{i}!(g_{i}-n_{i})!}\ .

NTCIR12-MathWiki-18rate: 0

( p n ) = p ! n ! ( p - n ) ! . binomial p n p n p n {\left({{p}\atop{n}}\right)}=\frac{p!}{n!(p-n)!}.

NTCIR12-MathWiki-18rate: 1

( n k ) = n ! k ! ( n - k ) ! . binomial n k n k n k {n\choose k}=\frac{n!}{k!(n-k)!}.

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x k 1 x k 2 N exp ( - 1 2 i , j = 1 n A i j x i x j ) d n x = ( 2 π ) n det A 1 2 N N ! σ S 2 N ( A - 1 ) k σ ( 1 ) k σ ( 2 ) ( A - 1 ) k σ ( 2 N - 1 ) k σ ( 2 N ) superscript x subscript k 1 normal-⋯ superscript x subscript k 2 N 1 2 superscript subscript i j 1 n subscript A i j subscript x i subscript x j superscript d n x superscript 2 π n A 1 superscript 2 N N subscript σ subscript S 2 N superscript superscript A 1 subscript k σ 1 subscript k σ 2 normal-⋯ superscript superscript A 1 subscript k σ 2 N 1 subscript k σ 2 N \int x^{k_{1}}\cdots x^{k_{2N}}\,\exp\left(-\frac{1}{2}\sum_{i,j=1}^{n}A_{ij}x% _{i}x_{j}\right)\,d^{n}x=\sqrt{\frac{(2\pi)^{n}}{\det A}}\,\frac{1}{2^{N}N!}\,% \sum_{\sigma\in S_{2N}}(A^{-1})^{k_{\sigma(1)}k_{\sigma(2)}}\cdots(A^{-1})^{k_% {\sigma(2N-1)}k_{\sigma(2N)}}

NTCIR12-MathWiki-18rate: 1

( n k ) = n ! k ! ( n - k ) ! binomial n k n k n k {n\choose k}={n!\over k!(n-k)!}

NTCIR12-MathWiki-18rate: 1

= ( A ) n A ( B ) n B ( N ! n A ! n B ! c o m b i n a t i o n ) ( n A π n A / 2 ( n A / 2 ) ! ( 2 E A ) n A - 1 2 n A - s p h e r e ) ( n B π n B / 2 ( n B / 2 ) ! ( 2 E B ) n B - 1 2 n B - s p h e r e ) absent superscript subscript normal-ℓ A subscript n A superscript subscript normal-ℓ B subscript n B subscript normal-⏟ N subscript n A subscript n B c o m b i n a t i o n subscript normal-⏟ subscript n A superscript π subscript n A 2 subscript n A 2 superscript 2 subscript E A subscript n A 1 2 subscript n A s p h e r e subscript normal-⏟ subscript n B superscript π subscript n B 2 subscript n B 2 superscript 2 subscript E B subscript n B 1 2 subscript n B s p h e r e =\left(\ell_{A}\right)^{n_{A}}\left(\ell_{B}\right)^{n_{B}}\left(\underbrace{% \frac{N!}{n_{A}!n_{B}!}}_{combination}\right)\left(\underbrace{\frac{n_{A}\pi^% {n_{A}/2}}{(n_{A}/2)!}(2E_{A})^{\frac{n_{A}-1}{2}}}_{n_{A}-sphere}\right)\left% (\underbrace{\frac{n_{B}\pi^{n_{B}/2}}{(n_{B}/2)!}(2E_{B})^{\frac{n_{B}-1}{2}}% }_{n_{B}-sphere}\right)

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n ! l 1 ! l 2 ! l k ! i < j ( l i - l j ) n subscript l 1 subscript l 2 normal-⋯ subscript l k subscript product i j subscript l i subscript l j \frac{n!}{l_{1}!l_{2}!\cdots l_{k}!}\prod_{i<j}(l_{i}-l_{j})

NTCIR12-MathWiki-18rate: 0

( n m ) = n ! m ! ( n - m ) ! binomial n m n m n m {n\choose m}=\frac{n!}{m!\,(n-m)!}

NTCIR12-MathWiki-18rate: 1

| x 1 x 2 x N ; S = j n j ! N ! p | x p ( 1 ) | x p ( 2 ) | x p ( N ) ket subscript x 1 subscript x 2 normal-⋯ subscript x N S subscript product j subscript n j N subscript p ket subscript x p 1 ket subscript x p 2 normal-⋯ ket subscript x p N |x_{1}x_{2}\cdots x_{N};S\rangle=\frac{\prod_{j}n_{j}!}{N!}\sum_{p}|x_{p(1)}% \rangle|x_{p(2)}\rangle\cdots|x_{p(N)}\rangle

NTCIR12-MathWiki-18rate: 2

P ( k t = k ) = ( K k ) p t k ( 1 - p t ) K - k , k = 0 , 1 , , K . fragments P fragments normal-( subscript k t k normal-) binomial K k superscript subscript p t k superscript fragments normal-( 1 subscript p t normal-) K k normal-, k 0 normal-, 1 normal-, normal-… normal-, K normal-. P(k_{t}=k)={K\choose k}p_{t}^{k}(1-p_{t})^{K-k},\,\quad\quad k=0,1,\dots,K.

NTCIR12-MathWiki-18rate: 0

μ ( σ , τ ) = ( - 1 ) n - r ( 2 ! ) r 3 ( 3 ! ) r 4 ( ( n - 1 ) ! ) r n μ σ τ superscript 1 n r superscript 2 subscript r 3 superscript 3 subscript r 4 normal-⋯ superscript n 1 subscript r n \mu(\sigma,\tau)=(-1)^{n-r}(2!)^{r_{3}}(3!)^{r_{4}}\cdots((n-1)!)^{r_{n}}

NTCIR12-MathWiki-18rate: 1

( n p ) = n ! p ! ( n - p ) ! binomial n p n p n p {n\choose p}=\frac{n!}{p!(n-p)!}

NTCIR12-MathWiki-18rate: 0

Ran e i ( T ) = Ker ( T - λ i ) ν i . Ran subscript e i T Ker superscript T subscript λ i subscript ν i \mathrm{Ran}\;e_{i}(T)=\mathrm{Ker}(T-\lambda_{i})^{\nu_{i}}.

NTCIR12-MathWiki-18rate: 0

2 N p K ( 1 - p ) N - K W . superscript 2 N superscript p K superscript 1 p N K W 2^{N}p^{K}(1-p)^{N-K}W\!.

NTCIR12-MathWiki-18rate: 1

Ω c o n f = N S ! N ! ( N S - N ) ! subscript normal-Ω c o n f subscript N S N subscript N S N \Omega_{conf}\,=\,\frac{N_{S}!}{N!(N_{S}-N)!}

NTCIR12-MathWiki-18rate: 3

Pr ( Y i = y i 𝐗 i ) = ( n i y i ) p i y i ( 1 - p i ) n i - y i = ( n i y i ) ( 1 1 + e - s y m b o l β 𝐗 i ) y i ( 1 - 1 1 + e - s y m b o l β 𝐗 i ) n i - y i Pr subscript Y i subscript y i subscript 𝐗 i binomial subscript n i subscript y i superscript subscript p i subscript y i superscript 1 subscript p i subscript n i subscript y i binomial subscript n i subscript y i superscript 1 1 superscript e normal-⋅ s y m b o l β subscript 𝐗 i subscript y i superscript 1 1 1 superscript e normal-⋅ s y m b o l β subscript 𝐗 i subscript n i subscript y i \operatorname{Pr}(Y_{i}=y_{i}\mid\mathbf{X}_{i})={n_{i}\choose y_{i}}p_{i}^{y_% {i}}(1-p_{i})^{n_{i}-y_{i}}={n_{i}\choose y_{i}}\left(\frac{1}{1+e^{-symbol% \beta\cdot\mathbf{X}_{i}}}\right)^{y_{i}}\left(1-\frac{1}{1+e^{-symbol\beta% \cdot\mathbf{X}_{i}}}\right)^{n_{i}-y_{i}}

NTCIR12-MathWiki-18rate: 1

( n k ) = n ! k ! ( n - k ) ! binomial n k n k n k {n\choose k}={n!\over k!(n-k)!}

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W = N ! i 1 N i ! W N subscript product i 1 subscript N i W=N!\prod_{i}\frac{1}{N_{i}!}

NTCIR12-MathWiki-18rate: 2

f ( k ) = ( n k ) p k ( 1 - p ) n - k f k binomial n k superscript p k superscript 1 p n k f(k)={n\choose k}p^{k}(1-p)^{n-k}

NTCIR12-MathWiki-18rate: 1

W = N ! N a ! ( N - N a ) ! × ( N - N a ) ! N b ! ( N - N a - N b ) ! × ( N - N a - N b ) ! N c ! ( N - N a - N b - N c ) ! × × ( N - - N l ) ! N k ! ( N - - N l - N k ) ! = = N ! N a ! N b ! N c ! N k ! ( N - - N l - N k ) ! W absent N subscript N a N subscript N a N subscript N a subscript N b N subscript N a subscript N b N subscript N a subscript N b subscript N c N subscript N a subscript N b subscript N c normal-… N normal-… subscript N l subscript N k N normal-… subscript N l subscript N k absent missing-subexpression missing-subexpression absent N subscript N a subscript N b subscript N c normal-… subscript N k N normal-… subscript N l subscript N k \begin{aligned}\displaystyle W&\displaystyle=\frac{N!}{N_{a}!(N-N_{a})!}\times% \frac{(N-N_{a})!}{N_{b}!(N-N_{a}-N_{b})!}~{}\times\frac{(N-N_{a}-N_{b})!}{N_{c% }!(N-N_{a}-N_{b}-N_{c})!}\times\ldots\times\frac{(N-\ldots-N_{l})!}{N_{k}!(N-% \ldots-N_{l}-N_{k})!}=\\ \\ &\displaystyle=\frac{N!}{N_{a}!N_{b}!N_{c}!\ldots N_{k}!(N-\ldots-N_{l}-N_{k})% !}\end{aligned}

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L * = n = 1 N P i Y i ( 1 - P i ) 1 - Y i superscript L superscript subscript product n 1 N superscript subscript P i subscript Y i superscript 1 subscript P i 1 subscript Y i L^{*}=\prod_{n=1}^{N}{P_{i}^{Y_{i}}}\left(1-P_{i}\right)^{1-Y_{i}}

NTCIR12-MathWiki-18rate: 0

d α i d x i α i x i β i = β i ! ( β i - α i ) ! x i β i - α i superscript d subscript α i d superscript subscript x i subscript α i superscript subscript x i subscript β i subscript β i subscript β i subscript α i superscript subscript x i subscript β i subscript α i \frac{d^{\alpha_{i}}}{dx_{i}^{\alpha_{i}}}x_{i}^{\beta_{i}}=\frac{\beta_{i}!}{% (\beta_{i}-\alpha_{i})!}x_{i}^{\beta_{i}-\alpha_{i}}

NTCIR12-MathWiki-18rate: 0

n ! k 1 ! k 2 ! k m - 1 ! K ! K ! k m ! k m + 1 ! = n ! k 1 ! k 2 ! k m + 1 ! . n subscript k 1 subscript k 2 normal-⋯ subscript k m 1 K K subscript k m subscript k m 1 n subscript k 1 subscript k 2 normal-⋯ subscript k m 1 \frac{n!}{k_{1}!k_{2}!\cdots k_{m-1}!K!}\frac{K!}{k_{m}!k_{m+1}!}=\frac{n!}{k_% {1}!k_{2}!\cdots k_{m+1}!}.

NTCIR12-MathWiki-18rate: 0

( N n 1 ) ( N - n 1 n 2 ) ( N - n 1 - n 2 n 3 ) = N ! ( N - n 1 ) ! n 1 ! ( N - n 1 ) ! ( N - n 1 - n 2 ) ! n 2 ! ( N - n 1 - n 2 ) ! ( N - n 1 - n 2 - n 3 ) ! n 3 ! . binomial N subscript n 1 binomial N subscript n 1 subscript n 2 binomial N subscript n 1 subscript n 2 subscript n 3 normal-… N N subscript n 1 subscript n 1 N subscript n 1 N subscript n 1 subscript n 2 subscript n 2 N subscript n 1 subscript n 2 N subscript n 1 subscript n 2 subscript n 3 subscript n 3 normal-… {N\choose n_{1}}{N-n_{1}\choose n_{2}}{N-n_{1}-n_{2}\choose n_{3}}...=\frac{N!% }{(N-n_{1})!n_{1}!}\frac{(N-n_{1})!}{(N-n_{1}-n_{2})!n_{2}!}\frac{(N-n_{1}-n_{% 2})!}{(N-n_{1}-n_{2}-n_{3})!n_{3}!}....

NTCIR12-MathWiki-18rate: 1

W ( n , N ) = ( N n ) = N ! n ! ( N - n ) ! W n N binomial N n N n N n W(n,N)={N\choose n}={{N!}\over{n!(N-n)!}}

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R 1 = 1 2 ! f ′′ ( ξ n ) ( α - x n ) 2 , subscript R 1 1 2 superscript f ′′ subscript ξ n superscript α subscript x n 2 R_{1}=\frac{1}{2!}f^{\prime\prime}(\xi_{n})(\alpha-x_{n})^{2}\,,

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n ! ( k - 1 ) ! ( n - k ) ! u k - 1 ( 1 - u ) n - k d u + O ( d u 2 ) , n k 1 n k superscript u k 1 superscript 1 u n k d u O d superscript u 2 {n!\over(k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}\,du+O(du^{2}),

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f X ( k ) ( x ) = n ! ( k - 1 ) ! ( n - k ) ! [ F X ( x ) ] k - 1 [ 1 - F X ( x ) ] n - k f X ( x ) subscript f subscript X k x n k 1 n k superscript delimited-[] subscript F X x k 1 superscript delimited-[] 1 subscript F X x n k subscript f X x f_{X_{(k)}}(x)=\frac{n!}{(k-1)!(n-k)!}[F_{X}(x)]^{k-1}[1-F_{X}(x)]^{n-k}f_{X}(x)

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P - Q i A i = O ( ( x - λ i ) ν i ) P subscript Q i subscript A i O superscript x subscript λ i subscript ν i P-Q_{i}A_{i}=O((x-\lambda_{i})^{\nu_{i}})\qquad

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Z = 1 N ! h 3 N exp [ - β H ( p 1 p N , x 1 x N ) ] d 3 p 1 d 3 p N d 3 x 1 d 3 x N Z 1 N superscript h 3 N β H subscript p 1 normal-⋯ subscript p N subscript x 1 normal-⋯ subscript x N superscript d 3 subscript p 1 normal-⋯ superscript d 3 subscript p N superscript d 3 subscript x 1 normal-⋯ superscript d 3 subscript x N Z=\frac{1}{N!h^{3N}}\int\,\exp[-\beta H(p_{1}\cdots p_{N},x_{1}\cdots x_{N})]% \;d^{3}p_{1}\cdots d^{3}p_{N}\,d^{3}x_{1}\cdots d^{3}x_{N}

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( n r ) = n ! r ! ( n - r ) ! binomial n r n r n r {\textstyle\left({{n}\atop{r}}\right)}=\tfrac{n!}{r!(n-r)!}

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α ! = Π i = 1 N ( α i ! ) α superscript subscript normal-Π i 1 N subscript α i \alpha!=\Pi_{i=1}^{N}(\alpha_{i}!)

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p n ( k ) = n ! ( n - k ) ! k ! p k ( 1 - p ) n - k . subscript p n k n n k k superscript p k superscript 1 p n k p_{n}(k)=\frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}.

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( n r ) = n ! r ! ( n - r ) ! = ( 52 5 ) = 52 ! 5 ! ( 52 - 5 ) ! = 2 , 598 , 960 formulae-sequence binomial n r n r n r binomial 52 5 52 5 52 5 2 598 960 {n\choose r}={{n!}\over{r!(n-r)!}}={52\choose 5}={{52!}\over{5!(52-5)!}}=2,598% ,960

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1 p ! Δ Q p f ( p , q , r ) = 1 ( n - p ) ! Δ P n - p f ( a , b , c ) . 1 p superscript subscript normal-Δ Q p f p q r 1 n p superscript subscript normal-Δ P n p f a b c \frac{1}{p!}\Delta_{Q}^{p}f(p,q,r)=\frac{1}{(n-p)!}\Delta_{P}^{n-p}f(a,b,c).

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1 N log W = 1 N log N ! n 1 ! n 2 ! n m ! = 1 N log N ! ( N p 1 ) ! ( N p 2 ) ! ( N p m ) ! = 1 N ( log N ! - i = 1 m log ( ( N p i ) ! ) ) . 1 N W 1 N N subscript n 1 subscript n 2 normal-⋯ subscript n m missing-subexpression missing-subexpression missing-subexpression missing-subexpression 1 N N N subscript p 1 N subscript p 2 normal-⋯ N subscript p m missing-subexpression missing-subexpression missing-subexpression missing-subexpression 1 N N superscript subscript i 1 m N subscript p i \begin{array}[]{rcl}\frac{1}{N}\log W&=&\frac{1}{N}\log\frac{N!}{n_{1}!\,n_{2}% !\,\cdots\,n_{m}!}\\ \\ &=&\frac{1}{N}\log\frac{N!}{(Np_{1})!\,(Np_{2})!\,\cdots\,(Np_{m})!}\\ \\ &=&\frac{1}{N}\left(\log N!-\sum_{i=1}^{m}\log((Np_{i})!)\right).\end{array}

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I B = log ( N ! ) - i = 1 K ( log ( n i ! ) ) N subscript I B N superscript subscript i 1 K subscript n i N I_{B}=\frac{\log(N!)-\sum_{i=1}^{K}(\log(n_{i}!))}{N}

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α = 4 p ( 1 - p ) . α 4 p 1 p \alpha=4p(1-p).\!

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C 0 N = ( 1 + r ) - N n = 0 N N ! n ! ( N - n ) ! q n ( 1 - q ) N - n [ S 0 ( 1 + b ) n ( 1 + a ) N - n - K ] + superscript subscript C 0 N superscript 1 r N superscript subscript n 0 N N n N n superscript q n superscript 1 q N n superscript delimited-[] subscript S 0 superscript 1 b n superscript 1 a N n K C_{0}^{N}=(1+r)^{-N}\sum_{n=0}^{N}\frac{N!}{n!(N-n)!}q^{n}{(1-q)}^{N-n}{[S_{0}% {(1+b)}^{n}{(1+a)}^{N-n}-K]}^{+}

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| C k | c k ! n ! ( k - n ) ! subscript C k superscript c k n k n |C_{k}|\leq c^{\frac{k!}{n!(k-n)!}}

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n ! ( n - r ) ! r ! μ r ( 1 - μ ) n - r e - λ λ r r ! . normal-→ n n r r superscript μ r superscript 1 μ n r superscript e λ superscript λ r r \frac{n!}{(n-r)!r!}\mu^{r}(1-\mu)^{n-r}\rightarrow e^{-\lambda}\frac{\lambda^{% r}}{r!}.

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f ( p ) = ( n + 1 ) ! s ! ( n - s ) ! p s ( 1 - p ) n - s . f p n 1 s n s superscript p s superscript 1 p n s f(p)={(n+1)!\over s!(n-s)!}p^{s}(1-p)^{n-s}.

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S k = ( m k ) ( m - k ) ! m ! = 1 k ! . subscript S k binomial m k m k m 1 k S_{k}={\left({{m}\atop{k}}\right)}\frac{(m-k)!}{m!}=\frac{1}{k!}.

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L ( θ ; A , B ) = ( A + B ) ! A ! B ! θ A ( 1 - θ ) B , L θ A B A B A B superscript θ A superscript 1 θ B L(\theta;A,B)=\frac{(A+B)!}{A!B!}\theta^{A}(1-\theta)^{B},

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P - m = ( - 1 ) m ( - m ) ! ( + m ) ! P m superscript subscript P normal-ℓ m superscript 1 m normal-ℓ m normal-ℓ m superscript subscript P normal-ℓ m P_{\ell}^{-m}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^{m}

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P i x = N ! n x ! ( N - n x ) ! p x n x ( 1 - p x ) N - n x superscript subscript P i x N subscript n x N subscript n x superscript subscript p x subscript n x superscript 1 subscript p x N subscript n x P_{i}^{x}=\frac{N!}{n_{x}!(N-n_{x})!}p_{x}^{n_{x}}(1-p_{x})^{N-n_{x}}

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e - n λ λ ( x 1 + x 2 + + x n ) 1 x 1 ! x 2 ! x n ! normal-⋅ superscript e n λ superscript λ subscript x 1 subscript x 2 normal-⋯ subscript x n 1 subscript x 1 subscript x 2 normal-⋯ subscript x n e^{-n\lambda}\lambda^{(x_{1}+x_{2}+\cdots+x_{n})}\cdot{1\over x_{1}!x_{2}!% \cdots x_{n}!}\,

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s 2 = n p ( 1 - p ) superscript s 2 n p 1 p s^{2}=np(1-p)

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s 2 = n p ( 1 - p ) superscript s 2 n p 1 p s^{2}=np(1-p)

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\cdotsymbol τ s i j x i = μ [ x i ( u i x j + u j x i ) ] + λ [ x i ( u k x k ) ] δ i j = μ 2 u i x i x j + μ 2 u j x i x i + λ 2 u k x k x j = ( μ + λ ) 2 u i x i x j + μ 2 u j x i 2 ( μ + λ ) ( 𝐮 ) + μ 2 𝐮 . normal-∇ \cdotsymbol τ continued-fraction subscript s i j subscript x i absent μ delimited-[] continued-fraction subscript x i continued-fraction subscript u i subscript x j continued-fraction subscript u j subscript x i λ delimited-[] continued-fraction subscript x i continued-fraction subscript u k subscript x k subscript δ i j missing-subexpression absent μ continued-fraction superscript 2 subscript u i subscript x i subscript x j μ continued-fraction superscript 2 subscript u j subscript x i subscript x i λ continued-fraction superscript 2 subscript u k subscript x k subscript x j missing-subexpression absent μ λ continued-fraction superscript 2 subscript u i subscript x i subscript x j μ continued-fraction superscript 2 subscript u j superscript subscript x i 2 missing-subexpression absent μ λ normal-∇ normal-⋅ normal-∇ 𝐮 μ superscript normal-∇ 2 𝐮 \begin{aligned}\displaystyle\nabla\cdotsymbol{\tau}\equiv\cfrac{\partial s_{ij% }}{\partial x_{i}}&\displaystyle=\mu\left[\cfrac{\partial}{\partial x_{i}}% \left(\cfrac{\partial u_{i}}{\partial x_{j}}+\cfrac{\partial u_{j}}{\partial x% _{i}}\right)\right]+\lambda~{}\left[\cfrac{\partial}{\partial x_{i}}\left(% \cfrac{\partial u_{k}}{\partial x_{k}}\right)\right]\delta_{ij}\\ &\displaystyle=\mu~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}\partial x_{j}}+% \mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}\partial x_{i}}+\lambda~{}\cfrac% {\partial^{2}u_{k}}{\partial x_{k}\partial x_{j}}\\ &\displaystyle=(\mu+\lambda)~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}% \partial x_{j}}+\mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}^{2}}\\ &\displaystyle\equiv(\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})+\mu~{}\nabla% ^{2}\mathbf{u}~{}.\end{aligned}

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2 ρ t 2 - c 0 2 2 ρ = 2 T i j x i x j , ( * ) superscript 2 ρ superscript t 2 subscript superscript c 2 0 superscript normal-∇ 2 ρ superscript 2 subscript T i j subscript x i subscript x j \frac{\partial^{2}\rho}{\partial t^{2}}-c^{2}_{0}\nabla^{2}\rho=\frac{\partial% ^{2}T_{ij}}{\partial x_{i}\partial x_{j}},\quad(*)

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1 c 0 2 2 p t 2 - 2 p = ρ 0 2 T ^ i j x i x j , where T ^ i j = v i v j . formulae-sequence 1 superscript subscript c 0 2 superscript 2 p superscript t 2 superscript normal-∇ 2 p subscript ρ 0 superscript 2 subscript normal-^ T i j subscript x i subscript x j where subscript normal-^ T i j subscript v i subscript v j \frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{\partial t^{2}}-\nabla^{2}p=\rho_{0}% \frac{\partial^{2}\hat{T}_{ij}}{\partial x_{i}\partial x_{j}},\quad\,\text{% where}\quad\hat{T}_{ij}=v_{i}v_{j}.

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𝕊 = ( S x x S x y S y x S y y ) = 𝕀 ( c g c p - 1 2 ) E + 1 k 2 ( k x k x k x k y k y k x k y k y ) c g c p E , 𝕀 = ( 1 0 0 1 ) and s y m b o l U = ( U x x U y x U x y U y y ) , 𝕊 absent subscript S x x subscript S x y subscript S y x subscript S y y 𝕀 subscript c g subscript c p 1 2 E 1 superscript k 2 subscript k x subscript k x subscript k x subscript k y subscript k y subscript k x subscript k y subscript k y subscript c g subscript c p E 𝕀 absent 1 0 0 1 and normal-∇ s y m b o l U absent subscript U x x subscript U y x subscript U x y subscript U y y \begin{aligned}\displaystyle\mathbb{S}&\displaystyle=\,\begin{pmatrix}S_{xx}&S% _{xy}\\ S_{yx}&S_{yy}\end{pmatrix}\,=\,\mathbb{I}\,\left(\frac{c_{g}}{c_{p}}-\frac{1}{% 2}\right)\,E\,+\,\frac{1}{k^{2}}\,\begin{pmatrix}k_{x}\,k_{x}&k_{x}\,k_{y}\\ k_{y}\,k_{x}&k_{y}\,k_{y}\end{pmatrix}\,\frac{c_{g}}{c_{p}}\,E,\\ \displaystyle\mathbb{I}&\displaystyle=\,\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\quad\,\text{and}\\ \displaystyle\nabla symbol{U}&\displaystyle=\,\begin{pmatrix}\displaystyle% \frac{\partial U_{x}}{\partial x}&\displaystyle\frac{\partial U_{y}}{\partial x% }\\ \displaystyle\frac{\partial U_{x}}{\partial y}&\displaystyle\frac{\partial U_{% y}}{\partial y}\end{pmatrix},\end{aligned}

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H i j = [ 2 V i j x i x j 2 V i j x i y j 2 V i j x i z j 2 V i j y i x j 2 V i j y i y j 2 V i j y i z j 2 V i j z i x j 2 V i j z i y j 2 V i j z i z j ] subscript H i j superscript 2 subscript V i j subscript x i subscript x j superscript 2 subscript V i j subscript x i subscript y j superscript 2 subscript V i j subscript x i subscript z j superscript 2 subscript V i j subscript y i subscript x j superscript 2 subscript V i j subscript y i subscript y j superscript 2 subscript V i j subscript y i subscript z j superscript 2 subscript V i j subscript z i subscript x j superscript 2 subscript V i j subscript z i subscript y j superscript 2 subscript V i j subscript z i subscript z j H_{ij}=\begin{bmatrix}{\partial^{2}V_{ij}\over\partial x_{i}\partial x_{j}}&{% \partial^{2}V_{ij}\over\partial x_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over% \partial x_{i}\partial z_{j}}\\ {\partial^{2}V_{ij}\over\partial y_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over% \partial y_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial y_{i}\partial z% _{j}}\\ {\partial^{2}V_{ij}\over\partial z_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over% \partial z_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial z_{i}\partial z% _{j}}\end{bmatrix}

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v x c A x + v y c A y = D A B 2 c A y 2 subscript v x subscript c A x subscript v y subscript c A y subscript D A B superscript 2 subscript c A superscript y 2 v_{x}{\partial c_{A}\over\partial x}+v_{y}{\partial c_{A}\over\partial y}=D_{% AB}{\partial^{2}c_{A}\over\partial y^{2}}

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X = | x 11 x 12 x 13 x 1 n x 12 x 22 x 23 x 2 n x 13 x 23 x 33 x 3 n x 1 n x 2 n x 3 n x n n | , D = | 2 x 11 x 12 x 13 x 1 n x 12 2 x 22 x 23 x 2 n x 13 x 23 2 x 33 x 3 n x 1 n x 2 n x 3 n 2 x n n | formulae-sequence X subscript x 11 subscript x 12 subscript x 13 normal-⋯ subscript x 1 n subscript x 12 subscript x 22 subscript x 23 normal-⋯ subscript x 2 n subscript x 13 subscript x 23 subscript x 33 normal-⋯ subscript x 3 n normal-⋮ normal-⋮ normal-⋮ normal-⋱ normal-⋮ subscript x 1 n subscript x 2 n subscript x 3 n normal-⋯ subscript x n n D 2 subscript x 11 subscript x 12 subscript x 13 normal-⋯ subscript x 1 n subscript x 12 2 subscript x 22 subscript x 23 normal-⋯ subscript x 2 n subscript x 13 subscript x 23 2 subscript x 33 normal-⋯ subscript x 3 n normal-⋮ normal-⋮ normal-⋮ normal-⋱ normal-⋮ subscript x 1 n subscript x 2 n subscript x 3 n normal-⋯ 2 subscript x n n X=\begin{vmatrix}x_{11}&x_{12}&x_{13}&\cdots&x_{1n}\\ x_{12}&x_{22}&x_{23}&\cdots&x_{2n}\\ x_{13}&x_{23}&x_{33}&\cdots&x_{3n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{1n}&x_{2n}&x_{3n}&\cdots&x_{nn}\end{vmatrix},D=\begin{vmatrix}2\frac{% \partial}{\partial x_{11}}&\frac{\partial}{\partial x_{12}}&\frac{\partial}{% \partial x_{13}}&\cdots&\frac{\partial}{\partial x_{1n}}\\ \frac{\partial}{\partial x_{12}}&2\frac{\partial}{\partial x_{22}}&\frac{% \partial}{\partial x_{23}}&\cdots&\frac{\partial}{\partial x_{2n}}\\ \frac{\partial}{\partial x_{13}}&\frac{\partial}{\partial x_{23}}&2\frac{% \partial}{\partial x_{33}}&\cdots&\frac{\partial}{\partial x_{3n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{1n}}&\frac{\partial}{\partial x_{2n}}&\frac{% \partial}{\partial x_{3n}}&\cdots&2\frac{\partial}{\partial x_{nn}}\end{vmatrix}

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σ i j = - ( p 0 0 0 p 0 0 0 p ) + μ ( 2 u x u y + v x u z + w x v x + u y 2 v y v z + w y w x + u z w y + v z 2 w z ) = - p I + μ ( 𝐯 + ( 𝐯 ) T ) subscript σ i j absent p 0 0 0 p 0 0 0 p μ 2 u x u y v x u z w x v x u y 2 v y v z w y w x u z w y v z 2 w z missing-subexpression absent p I μ normal-∇ 𝐯 superscript normal-∇ 𝐯 T \begin{aligned}\displaystyle\sigma_{ij}&\displaystyle=-\begin{pmatrix}p&0&0\\ 0&p&0\\ 0&0&p\end{pmatrix}+\mu\begin{pmatrix}2\frac{\partial u}{\partial x}&\frac{% \partial u}{\partial y}+\frac{\partial v}{\partial x}&\frac{\partial u}{% \partial z}+\frac{\partial w}{\partial x}\\ \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}&2\frac{\partial v}% {\partial y}&\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\\ \frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}&\frac{\partial w}{% \partial y}+\frac{\partial v}{\partial z}&2\frac{\partial w}{\partial z}\end{% pmatrix}\\ &\displaystyle=-pI+\mu(\nabla\mathbf{v}+(\nabla\mathbf{v})^{T})\end{aligned}

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( x ¯ 1 x ¯ 2 x ¯ 3 ) = ( x ¯ 1 x 1 x ¯ 1 x 2 x ¯ 1 x 3 x ¯ 2 x 1 x ¯ 2 x 2 x ¯ 2 x 3 x ¯ 3 x 1 x ¯ 3 x 2 x ¯ 3 x 3 ) ( x 1 x 2 x 3 ) subscript normal-¯ x 1 subscript normal-¯ x 2 subscript normal-¯ x 3 subscript normal-¯ x 1 subscript x 1 subscript normal-¯ x 1 subscript x 2 subscript normal-¯ x 1 subscript x 3 subscript normal-¯ x 2 subscript x 1 subscript normal-¯ x 2 subscript x 2 subscript normal-¯ x 2 subscript x 3 subscript normal-¯ x 3 subscript x 1 subscript normal-¯ x 3 subscript x 2 subscript normal-¯ x 3 subscript x 3 subscript x 1 subscript x 2 subscript x 3 \begin{pmatrix}\bar{x}_{1}\\ \bar{x}_{2}\\ \bar{x}_{3}\end{pmatrix}=\begin{pmatrix}\frac{\partial\bar{x}_{1}}{\partial x_% {1}}&\frac{\partial\bar{x}_{1}}{\partial x_{2}}&\frac{\partial\bar{x}_{1}}{% \partial x_{3}}\\ \frac{\partial\bar{x}_{2}}{\partial x_{1}}&\frac{\partial\bar{x}_{2}}{\partial x% _{2}}&\frac{\partial\bar{x}_{2}}{\partial x_{3}}\\ \frac{\partial\bar{x}_{3}}{\partial x_{1}}&\frac{\partial\bar{x}_{3}}{\partial x% _{2}}&\frac{\partial\bar{x}_{3}}{\partial x_{3}}\end{pmatrix}\begin{pmatrix}x_% {1}\\ x_{2}\\ x_{3}\end{pmatrix}

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D f ( x , y ) = [ u x u y v x v y ] D f x y u x u y v x v y Df(x,y)=\begin{bmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{% \partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{bmatrix}

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Ω = | x 11 x 1 n x n 1 x n n | . normal-Ω subscript x 11 normal-⋯ subscript x 1 n normal-⋮ normal-⋱ normal-⋮ subscript x n 1 normal-⋯ subscript x n n \Omega=\begin{vmatrix}\frac{\partial}{\partial x_{11}}&\cdots&\frac{\partial}{% \partial x_{1n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{n1}}&\cdots&\frac{\partial}{\partial x_{nn}}\end{% vmatrix}.

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2 y x i x j = k ( y u k 2 u k x i x j ) + k , ( 2 y u k u u k x i u x j ) . superscript 2 y subscript x i subscript x j subscript k y subscript u k superscript 2 subscript u k subscript x i subscript x j subscript k normal-ℓ superscript 2 y subscript u k subscript u normal-ℓ subscript u k subscript x i subscript u normal-ℓ subscript x j \frac{\partial^{2}y}{\partial x_{i}\partial x_{j}}=\sum_{k}\left(\frac{% \partial y}{\partial u_{k}}\frac{\partial^{2}u_{k}}{\partial x_{i}\partial x_{% j}}\right)+\sum_{k,\ell}\left(\frac{\partial^{2}y}{\partial u_{k}\partial u_{% \ell}}\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{\ell}}{\partial x% _{j}}\right).

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𝐉 = ( σ 1 λ 1 σ 1 λ 2 σ 1 λ n σ 2 λ 1 σ 2 λ 2 σ 2 λ n σ n λ 1 σ n λ 2 σ n λ n ) . 𝐉 subscript σ 1 subscript λ 1 subscript σ 1 subscript λ 2 normal-… subscript σ 1 subscript λ n subscript σ 2 subscript λ 1 subscript σ 2 subscript λ 2 normal-… subscript σ 2 subscript λ n normal-⋮ normal-⋮ normal-⋱ normal-⋮ subscript σ n subscript λ 1 subscript σ n subscript λ 2 normal-… subscript σ n subscript λ n \mathbf{J}=\left(\begin{array}[]{cccc}\frac{\partial\sigma_{1}}{\partial% \lambda_{1}}&\frac{\partial\sigma_{1}}{\partial\lambda_{2}}&\dots&\frac{% \partial\sigma_{1}}{\partial\lambda_{n}}\\ \frac{\partial\sigma_{2}}{\partial\lambda_{1}}&\frac{\partial\sigma_{2}}{% \partial\lambda_{2}}&\dots&\frac{\partial\sigma_{2}}{\partial\lambda_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\sigma_{n}}{\partial\lambda_{1}}&\frac{\partial\sigma_{n}}{% \partial\lambda_{2}}&\dots&\frac{\partial\sigma_{n}}{\partial\lambda_{n}}\end{% array}\right).

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𝐉 = [ x 1 q 1 x 1 q 2 x 1 q 3 x 2 q 1 x 2 q 2 x 2 q 3 x 3 q 1 x 3 q 2 x 3 q 3 ] , 𝐉 - 1 = [ q 1 x 1 q 1 x 2 q 1 x 3 q 2 x 1 q 2 x 2 q 2 x 3 q 3 x 1 q 3 x 2 q 3 x 3 ] formulae-sequence 𝐉 continued-fraction subscript x 1 superscript q 1 continued-fraction subscript x 1 superscript q 2 continued-fraction subscript x 1 superscript q 3 continued-fraction subscript x 2 superscript q 1 continued-fraction subscript x 2 superscript q 2 continued-fraction subscript x 2 superscript q 3 continued-fraction subscript x 3 superscript q 1 continued-fraction subscript x 3 superscript q 2 continued-fraction subscript x 3 superscript q 3 superscript 𝐉 1 continued-fraction superscript q 1 subscript x 1 continued-fraction superscript q 1 subscript x 2 continued-fraction superscript q 1 subscript x 3 continued-fraction superscript q 2 subscript x 1 continued-fraction superscript q 2 subscript x 2 continued-fraction superscript q 2 subscript x 3 continued-fraction superscript q 3 subscript x 1 continued-fraction superscript q 3 subscript x 2 continued-fraction superscript q 3 subscript x 3 \mathbf{J}=\begin{bmatrix}\cfrac{\partial x_{1}}{\partial q^{1}}&\cfrac{% \partial x_{1}}{\partial q^{2}}&\cfrac{\partial x_{1}}{\partial q^{3}}\\ \cfrac{\partial x_{2}}{\partial q^{1}}&\cfrac{\partial x_{2}}{\partial q^{2}}&% \cfrac{\partial x_{2}}{\partial q^{3}}\\ \cfrac{\partial x_{3}}{\partial q^{1}}&\cfrac{\partial x_{3}}{\partial q^{2}}&% \cfrac{\partial x_{3}}{\partial q^{3}}\\ \end{bmatrix},\quad\mathbf{J}^{-1}=\begin{bmatrix}\cfrac{\partial q^{1}}{% \partial x_{1}}&\cfrac{\partial q^{1}}{\partial x_{2}}&\cfrac{\partial q^{1}}{% \partial x_{3}}\\ \cfrac{\partial q^{2}}{\partial x_{1}}&\cfrac{\partial q^{2}}{\partial x_{2}}&% \cfrac{\partial q^{2}}{\partial x_{3}}\\ \cfrac{\partial q^{3}}{\partial x_{1}}&\cfrac{\partial q^{3}}{\partial x_{2}}&% \cfrac{\partial q^{3}}{\partial x_{3}}\\ \end{bmatrix}

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V [ u i t + u i u j x j ] d V = V [ - P x i + ν 2 u i x j x j + f i ] d V subscript triple-integral V delimited-[] subscript u i t subscript u i subscript u j subscript x j d V subscript triple-integral V delimited-[] P subscript x i ν superscript 2 subscript u i subscript x j subscript x j subscript f i d V \iiint_{V}\left[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{% \partial x_{j}}\right]dV=\iiint_{V}\left[-\frac{\partial P}{\partial x_{i}}+% \nu\frac{\partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}+f_{i}\right]dV

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ε = [ v 1 S 1 v 1 S m v n S 1 v n S m ] . ε subscript v 1 subscript S 1 normal-⋯ subscript v 1 subscript S m normal-⋮ normal-⋱ normal-⋮ subscript v n subscript S 1 normal-⋯ subscript v n subscript S m \mathbf{\varepsilon}=\begin{bmatrix}\dfrac{\partial v_{1}}{\partial S_{1}}&% \cdots&\dfrac{\partial v_{1}}{\partial S_{m}}\\ \vdots&\ddots&\vdots\\ \dfrac{\partial v_{n}}{\partial S_{1}}&\cdots&\dfrac{\partial v_{n}}{\partial S% _{m}}\end{bmatrix}.

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V i j = 2 V x i x j . subscript V i j superscript 2 V subscript x i subscript x j V_{ij}=\frac{\partial^{2}V}{\partial x_{i}\partial x_{j}}.

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Σ θ m = [ Σ 1 , 1 θ m Σ 1 , 2 θ m Σ 1 , N θ m Σ 2 , 1 θ m Σ 2 , 2 θ m Σ 2 , N θ m Σ N , 1 θ m Σ N , 2 θ m Σ N , N θ m ] . normal-Σ subscript θ m subscript normal-Σ 1 1 subscript θ m subscript normal-Σ 1 2 subscript θ m normal-⋯ subscript normal-Σ 1 N subscript θ m absent subscript normal-Σ 2 1 subscript θ m subscript normal-Σ 2 2 subscript θ m normal-⋯ subscript normal-Σ 2 N subscript θ m absent normal-⋮ normal-⋮ normal-⋱ normal-⋮ absent subscript normal-Σ N 1 subscript θ m subscript normal-Σ N 2 subscript θ m normal-⋯ subscript normal-Σ N N subscript θ m \frac{\partial\Sigma}{\partial\theta_{m}}=\begin{bmatrix}\frac{\partial\Sigma_% {1,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{1,2}}{\partial\theta_{m}}&% \cdots&\frac{\partial\Sigma_{1,N}}{\partial\theta_{m}}\\ \\ \frac{\partial\Sigma_{2,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{2,2}}{% \partial\theta_{m}}&\cdots&\frac{\partial\Sigma_{2,N}}{\partial\theta_{m}}\\ \\ \vdots&\vdots&\ddots&\vdots\\ \\ \frac{\partial\Sigma_{N,1}}{\partial\theta_{m}}&\frac{\partial\Sigma_{N,2}}{% \partial\theta_{m}}&\cdots&\frac{\partial\Sigma_{N,N}}{\partial\theta_{m}}\end% {bmatrix}.

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τ i j = μ ( v i x j + v j x i - 2 3 δ i j 𝐯 ) + κ δ i j 𝐯 subscript τ i j μ subscript v i subscript x j subscript v j subscript x i normal-⋅ 2 3 subscript δ i j normal-∇ 𝐯 normal-⋅ κ subscript δ i j normal-∇ 𝐯 \tau_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}% {\partial x_{i}}-\frac{2}{3}\delta_{ij}\nabla\cdot\mathbf{v}\right)+\kappa% \delta_{ij}\nabla\cdot\mathbf{v}

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[ x 1 x x 1 y x 1 z x 2 x x 2 y x 2 z x 3 x x 3 y x 3 z ] subscript x 1 x subscript x 1 y subscript x 1 z subscript x 2 x subscript x 2 y subscript x 2 z subscript x 3 x subscript x 3 y subscript x 3 z \begin{bmatrix}\dfrac{\partial x_{1}}{\partial x}&\dfrac{\partial x_{1}}{% \partial y}&\dfrac{\partial x_{1}}{\partial z}\\ \dfrac{\partial x_{2}}{\partial x}&\dfrac{\partial x_{2}}{\partial y}&\dfrac{% \partial x_{2}}{\partial z}\\ \dfrac{\partial x_{3}}{\partial x}&\dfrac{\partial x_{3}}{\partial y}&\dfrac{% \partial x_{3}}{\partial z}\end{bmatrix}

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E i j := 1 2 [ u i x j + u j x i + u k x i u k x j ] . assign subscript E i j 1 2 delimited-[] subscript u i subscript x j subscript u j subscript x i subscript u k subscript x i subscript u k subscript x j E_{ij}:=\frac{1}{2}\left[\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u% _{j}}{\partial x_{i}}+\frac{\partial u_{k}}{\partial x_{i}}\,\frac{\partial u_% {k}}{\partial x_{j}}\right]\,.

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2 S β j β k superscript 2 S subscript β j subscript β k \frac{\partial^{2}S}{\partial\beta_{j}\partial\beta_{k}}

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A i j = β i θ j = δ i j - α i θ j = δ i j - 2 ψ θ i θ j subscript A i j subscript β i subscript θ j subscript δ i j subscript α i subscript θ j subscript δ i j superscript 2 ψ subscript θ i subscript θ j A_{ij}=\frac{\partial\beta_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial% \alpha_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial^{2}\psi}{\partial% \theta_{i}\partial\theta_{j}}

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H ( f , g ) = [ 0 g x 1 g x 2 g x n g x 1 2 f x 1 2 2 f x 1 x 2 2 f x 1 x n g x 2 2 f x 2 x 1 2 f x 2 2 2 f x 2 x n g x n 2 f x n x 1 2 f x n x 2 2 f x n 2 ] H f g 0 g subscript x 1 g subscript x 2 normal-⋯ g subscript x n g subscript x 1 superscript 2 f superscript subscript x 1 2 superscript 2 f subscript x 1 subscript x 2 normal-⋯ superscript 2 f subscript x 1 subscript x n g subscript x 2 superscript 2 f subscript x 2 subscript x 1 superscript 2 f superscript subscript x 2 2 normal-⋯ superscript 2 f subscript x 2 subscript x n normal-⋮ normal-⋮ normal-⋮ normal-⋱ normal-⋮ g subscript x n superscript 2 f subscript x n subscript x 1 superscript 2 f subscript x n subscript x 2 normal-⋯ superscript 2 f superscript subscript x n 2 H(f,g)=\begin{bmatrix}0&\dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial g}{% \partial x_{2}}&\cdots&\dfrac{\partial g}{\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&% \dfrac{\partial^{2}f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{% 2}f}{\partial x_{1}\,\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{2}}&\dfrac{\partial^{2}f}{\partial x_{2}\,% \partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{2}^{2}}&\cdots&\dfrac{% \partial^{2}f}{\partial x_{2}\,\partial x_{n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial g}{\partial x_{n}}&\dfrac{\partial^{2}f}{\partial x_{n}\,% \partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{2}}&\cdots&% \dfrac{\partial^{2}f}{\partial x_{n}^{2}}\end{bmatrix}

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H = [ 2 f x 1 2 2 f x 1 x 2 2 f x 1 x n 2 f x 2 x 1 2 f x 2 2 2 f x 2 x n 2 f x n x 1 2 f x n x 2 2 f x n 2 ] . H superscript 2 f superscript subscript x 1 2 superscript 2 f subscript x 1 subscript x 2 normal-⋯ superscript 2 f subscript x 1 subscript x n superscript 2 f subscript x 2 subscript x 1 superscript 2 f superscript subscript x 2 2 normal-⋯ superscript 2 f subscript x 2 subscript x n normal-⋮ normal-⋮ normal-⋱ normal-⋮ superscript 2 f subscript x n subscript x 1 superscript 2 f subscript x n subscript x 2 normal-⋯ superscript 2 f superscript subscript x n 2 H=\begin{bmatrix}\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&\dfrac{\partial^{2}% f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{1}% \,\partial x_{n}}\\ \dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{1}}&\dfrac{\partial^{2}f}{% \partial x_{2}^{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{n% }}\\ \vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{1}}&\dfrac{\partial^{2}f}{% \partial x_{n}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{n}^{2% }}\end{bmatrix}.

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L u = a i j ( x ) 2 u x i x j + b i ( x ) u x i + c ( x ) u , x Ω . formulae-sequence L u subscript a i j x superscript 2 u subscript x i subscript x j subscript b i x u subscript x i c x u x normal-Ω Lu=a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}+b_{i}(x)\frac{% \partial u}{\partial x_{i}}+c(x)u,\qquad x\in\Omega.

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q i j = 2 V x i x j . subscript q i j superscript 2 V subscript x i subscript x j q_{ij}=\frac{\partial^{2}V}{\partial x_{i}\partial x_{j}}.

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2 ϵ y z 2 + 2 ϵ z y 2 = 2 2 ϵ y z y z superscript 2 subscript ϵ y superscript z 2 superscript 2 subscript ϵ z superscript y 2 2 superscript 2 subscript ϵ y z y z \frac{\partial^{2}\epsilon_{y}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial y^{2}}=2\frac{\partial^{2}\epsilon_{yz}}{\partial y\partial z}\,\!

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[ ε x x ε x y ε x z ε y x ε y y ε y z ε z x ε z y ε z z ] = [ u x x 1 2 ( u x y + u y x ) 1 2 ( u x z + u z x ) 1 2 ( u y x + u x y ) u y y 1 2 ( u y z + u z y ) 1 2 ( u z x + u x z ) 1 2 ( u z y + u y z ) u z z ] delimited-[] subscript ε x x subscript ε x y subscript ε x z subscript ε y x subscript ε y y subscript ε y z subscript ε z x subscript ε z y subscript ε z z delimited-[] subscript u x x 1 2 subscript u x y subscript u y x 1 2 subscript u x z subscript u z x 1 2 subscript u y x subscript u x y subscript u y y 1 2 subscript u y z subscript u z y 1 2 subscript u z x subscript u x z 1 2 subscript u z y subscript u y z subscript u z z \left[\begin{matrix}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz}\\ \end{matrix}\right]=\left[\begin{matrix}\frac{\partial u_{x}}{\partial x}&% \frac{1}{2}\left(\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{% \partial x}\right)&\frac{1}{2}\left(\frac{\partial u_{x}}{\partial z}+\frac{% \partial u_{z}}{\partial x}\right)\\ \frac{1}{2}\left(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{% \partial y}\right)&\frac{\partial u_{y}}{\partial y}&\frac{1}{2}\left(\frac{% \partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}\right)\\ \frac{1}{2}\left(\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{% \partial z}\right)&\frac{1}{2}\left(\frac{\partial u_{z}}{\partial y}+\frac{% \partial u_{y}}{\partial z}\right)&\frac{\partial u_{z}}{\partial z}\\ \end{matrix}\right]\,\!

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H ( f ) = [ 2 f x 2 2 f x y 2 f y x 2 f y 2 ] . H f superscript 2 f superscript x 2 superscript 2 f x y superscript 2 f y x superscript 2 f superscript y 2 H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{% \partial x\,\partial y}\\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{% 2}}\end{bmatrix}.

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y ˙ = y x j x ˙ j + 1 2 2 y x k x l g k m g m l . normal-˙ y y subscript x j subscript normal-˙ x j 1 2 superscript 2 y subscript x k subscript x l subscript g k m subscript g m l \dot{y}=\frac{\partial y}{\partial x_{j}}\dot{x}_{j}+\frac{1}{2}\frac{\partial% ^{2}y}{\partial x_{k}\,\partial x_{l}}g_{km}g_{ml}.

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J F = | f 1 X 1 f 1 X N f N X 1 f N X N | , subscript J F subscript f 1 subscript X 1 normal-⋯ subscript f 1 subscript X N normal-⋮ normal-⋱ normal-⋮ subscript f N subscript X 1 normal-⋯ subscript f N subscript X N J_{F}=\left|\begin{matrix}\frac{\partial f_{1}}{\partial X_{1}}&\cdots&\frac{% \partial f_{1}}{\partial X_{N}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{N}}{\partial X_{1}}&\cdots&\frac{\partial f_{N}}{\partial X_% {N}}\end{matrix}\right|,

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𝐉 𝐅 ( r , θ , φ ) = [ x r x θ x φ y r y θ y φ z r z θ z φ ] = [ sin θ cos φ r cos θ cos φ - r sin θ sin φ sin θ sin φ r cos θ sin φ r sin θ cos φ cos θ - r sin θ 0 ] . subscript 𝐉 𝐅 r θ φ x r x θ x φ y r y θ y φ z r z θ z φ θ φ r θ φ r θ φ θ φ r θ φ r θ φ θ r θ 0 \mathbf{J}_{\mathbf{F}}(r,\theta,\varphi)=\begin{bmatrix}\dfrac{\partial x}{% \partial r}&\dfrac{\partial x}{\partial\theta}&\dfrac{\partial x}{\partial% \varphi}\\ \dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial\theta}&\dfrac{% \partial y}{\partial\varphi}\\ \dfrac{\partial z}{\partial r}&\dfrac{\partial z}{\partial\theta}&\dfrac{% \partial z}{\partial\varphi}\end{bmatrix}=\begin{bmatrix}\sin\theta\cos\varphi% &r\cos\theta\cos\varphi&-r\sin\theta\sin\varphi\\ \sin\theta\sin\varphi&r\cos\theta\sin\varphi&r\sin\theta\cos\varphi\\ \cos\theta&-r\sin\theta&0\end{bmatrix}.

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𝐉 𝐅 ( x 1 , x 2 , x 3 ) = [ y 1 x 1 y 1 x 2 y 1 x 3 y 2 x 1 y 2 x 2 y 2 x 3 y 3 x 1 y 3 x 2 y 3 x 3 y 4 x 1 y 4 x 2 y 4 x 3 ] = [ 1 0 0 0 0 5 0 8 x 2 - 2 x 3 cos x 1 0 sin x 1 ] . subscript 𝐉 𝐅 subscript x 1 subscript x 2 subscript x 3 subscript y 1 subscript x 1 subscript y 1 subscript x 2 subscript y 1 subscript x 3 subscript y 2 subscript x 1 subscript y 2 subscript x 2 subscript y 2 subscript x 3 subscript y 3 subscript x 1 subscript y 3 subscript x 2 subscript y 3 subscript x 3 subscript y 4 subscript x 1 subscript y 4 subscript x 2 subscript y 4 subscript x 3 1 0 0 0 0 5 0 8 subscript x 2 2 subscript x 3 subscript x 1 0 subscript x 1 \mathbf{J}_{\mathbf{F}}(x_{1},x_{2},x_{3})=\begin{bmatrix}\dfrac{\partial y_{1% }}{\partial x_{1}}&\dfrac{\partial y_{1}}{\partial x_{2}}&\dfrac{\partial y_{1% }}{\partial x_{3}}\\ \dfrac{\partial y_{2}}{\partial x_{1}}&\dfrac{\partial y_{2}}{\partial x_{2}}&% \dfrac{\partial y_{2}}{\partial x_{3}}\\ \dfrac{\partial y_{3}}{\partial x_{1}}&\dfrac{\partial y_{3}}{\partial x_{2}}&% \dfrac{\partial y_{3}}{\partial x_{3}}\\ \dfrac{\partial y_{4}}{\partial x_{1}}&\dfrac{\partial y_{4}}{\partial x_{2}}&% \dfrac{\partial y_{4}}{\partial x_{3}}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&0&5\\ 0&8x_{2}&-2\\ x_{3}\cos x_{1}&0&\sin x_{1}\end{bmatrix}.

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2 u x i y j - 2 u y i x j = 0 superscript 2 u subscript x i subscript y j superscript 2 u subscript y i subscript x j 0 \frac{\partial^{2}u}{\partial x_{i}\partial y_{j}}-\frac{\partial^{2}u}{% \partial y_{i}\partial x_{j}}=0

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σ j k ϵ j k x 1 = σ j k 2 u j x 1 x k subscript σ j k continued-fraction subscript ϵ j k subscript x 1 subscript σ j k continued-fraction superscript 2 subscript u j subscript x 1 subscript x k \sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_{1}}=\sigma_{jk}\cfrac{% \partial^{2}u_{j}}{\partial x_{1}\partial x_{k}}

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δ = 1 4 π R s t | x y z x s y s z s x t y t z t | ( x 2 + y 2 + z 2 ) x 2 + y 2 + z 2 d s d t . δ 1 4 π subscript double-integral subscript R s t x y z x s y s z s x t y t z t superscript x 2 superscript y 2 superscript z 2 superscript x 2 superscript y 2 superscript z 2 d s d t \delta=\frac{1}{4\pi}\iint_{R_{st}}\frac{\begin{vmatrix}x&y&z\\ \dfrac{\partial x}{\partial s}&\dfrac{\partial y}{\partial s}&\dfrac{\partial z% }{\partial s}\\ \dfrac{\partial x}{\partial t}&\dfrac{\partial y}{\partial t}&\dfrac{\partial z% }{\partial t}\end{vmatrix}}{(x^{2}+y^{2}+z^{2})\sqrt{x^{2}+y^{2}+z^{2}}}dsdt.

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u i ¯ t + u j ¯ u i ¯ x j = - 1 ρ p ¯ x i + ν 2 u i ¯ x j x j - τ i j x j . normal-¯ subscript u i t normal-¯ subscript u j normal-¯ subscript u i subscript x j 1 ρ normal-¯ p subscript x i ν superscript 2 normal-¯ subscript u i subscript x j subscript x j subscript τ i j subscript x j \frac{\partial\bar{u_{i}}}{\partial t}+\bar{u_{j}}\frac{\partial\bar{u_{i}}}{% \partial x_{j}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i}}+\nu\frac% {\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}-\frac{\partial\tau_{ij% }}{\partial x_{j}}.

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2 ϵ y z 2 + 2 ϵ z y 2 = 2 2 ϵ y z y z superscript 2 subscript ϵ y superscript z 2 superscript 2 subscript ϵ z superscript y 2 2 superscript 2 subscript ϵ y z y z \frac{\partial^{2}\epsilon_{y}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}% }{\partial y^{2}}=2\frac{\partial^{2}\epsilon_{yz}}{\partial y\partial z}\,\!

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𝐀 = [ F 1 ( q ) p 1 F 1 ( q ) p 2 F 1 ( q ) p n F 2 ( q ) p 1 F 2 ( q ) p n - 1 F 2 ( q ) p n F j ( q ) p i F m ( q ) p 1 F m ( q ) p 2 F m ( q ) p n ] 𝐀 subscript F 1 normal-→ q subscript p 1 subscript F 1 normal-→ q subscript p 2 normal-⋯ subscript F 1 normal-→ q subscript p n subscript F 2 normal-→ q subscript p 1 normal-⋯ subscript F 2 normal-→ q subscript p n 1 subscript F 2 normal-→ q subscript p n normal-⋮ subscript F j normal-→ q subscript p i normal-⋮ normal-⋮ subscript F m normal-→ q subscript p 1 subscript F m normal-→ q subscript p 2 normal-⋯ subscript F m normal-→ q subscript p n \mathbf{A}=\begin{bmatrix}\frac{\partial F_{1}(\vec{q})}{\partial p_{1}}&\frac% {\partial F_{1}(\vec{q})}{\partial p_{2}}&\cdots&\frac{\partial F_{1}(\vec{q})% }{\partial p_{n}}\\ \frac{\partial F_{2}(\vec{q})}{\partial p_{1}}&\cdots&\frac{\partial F_{2}(% \vec{q})}{\partial p_{n-1}}&\frac{\partial F_{2}(\vec{q})}{\partial p_{n}}\\ \vdots&\frac{\partial F_{j}(\vec{q})}{\partial p_{i}}&\vdots&\vdots\\ \frac{\partial F_{m}(\vec{q})}{\partial p_{1}}&\frac{\partial F_{m}(\vec{q})}{% \partial p_{2}}&\cdots&\frac{\partial F_{m}(\vec{q})}{\partial p_{n}}\\ \end{bmatrix}\!

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ϕ ( 𝐗 ) 𝐗 = [ ϕ x 1 , 1 ϕ x 1 , q ϕ x n , 1 ϕ x n , q ] ϕ 𝐗 𝐗 ϕ subscript x 1 1 normal-⋯ ϕ subscript x 1 q normal-⋮ normal-⋱ normal-⋮ ϕ subscript x n 1 normal-⋯ ϕ subscript x n q \frac{\partial\mathbf{\phi}(\mathbf{X})}{\partial\mathbf{X}}=\begin{bmatrix}% \frac{\partial\mathbf{\phi}}{\partial x_{1,1}}&\cdots&\frac{\partial\mathbf{% \phi}}{\partial x_{1,q}}\\ \vdots&\ddots&\vdots\\ \frac{\partial\mathbf{\phi}}{\partial x_{n,1}}&\cdots&\frac{\partial\mathbf{% \phi}}{\partial x_{n,q}}\\ \end{bmatrix}

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y 𝐗 = [ y x 11 y x 12 y x 1 q y x 21 y x 22 y x 2 q y x p 1 y x p 2 y x p q ] . y 𝐗 y subscript x 11 y subscript x 12 normal-⋯ y subscript x 1 q y subscript x 21 y subscript x 22 normal-⋯ y subscript x 2 q normal-⋮ normal-⋮ normal-⋱ normal-⋮ y subscript x p 1 y subscript x p 2 normal-⋯ y subscript x p q \frac{\partial y}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial y}{% \partial x_{11}}&\frac{\partial y}{\partial x_{12}}&\cdots&\frac{\partial y}{% \partial x_{1q}}\\ \frac{\partial y}{\partial x_{21}}&\frac{\partial y}{\partial x_{22}}&\cdots&% \frac{\partial y}{\partial x_{2q}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y}{\partial x_{p1}}&\frac{\partial y}{\partial x_{p2}}&\cdots&% \frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}.

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( 2 S y x ) = ( 2 S x y ) : ( 2 V y x ) = ( 2 V x y ) normal-: superscript 2 S y x superscript 2 S x y superscript 2 V y x superscript 2 V x y \left(\frac{\partial^{2}S}{\partial y\partial x}\right)=\left(\frac{\partial^{% 2}S}{\partial x\partial y}\right):\left(\frac{\partial^{2}V}{\partial y% \partial x}\right)=\left(\frac{\partial^{2}V}{\partial x\partial y}\right)

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Hess ( F ) = ( 2 F x 2 2 F x y 2 F x z 2 F x y 2 F y 2 2 F y z 2 F x z 2 F y z 2 F z 2 ) . Hess F superscript 2 F superscript x 2 superscript 2 F x y superscript 2 F x z superscript 2 F x y superscript 2 F superscript y 2 superscript 2 F y z superscript 2 F x z superscript 2 F y z superscript 2 F superscript z 2 \textstyle\mbox{Hess}~{}(F)=\begin{pmatrix}\frac{\partial^{2}F}{\partial x^{2}% }&\frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial x% \partial z}\\ \frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial y^{2}% }&\frac{\partial^{2}F}{\partial y\partial z}\\ \frac{\partial^{2}F}{\partial x\partial z}&\frac{\partial^{2}F}{\partial y% \partial z}&\frac{\partial^{2}F}{\partial z^{2}}\end{pmatrix}.

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2 S ( z ) z i z j | z = 0 = 2 h i j ( 0 ) ; evaluated-at superscript 2 S z subscript z i subscript z j z 0 2 subscript h i j 0 \left.\frac{\partial^{2}S(z)}{\partial z_{i}\partial z_{j}}\right|_{z=0}=2h_{% ij}(0);

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J = [ u u u v v u v v ] . J u superscript u normal-′ u superscript v normal-′ v superscript u normal-′ v superscript v normal-′ J=\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{% \partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}% \end{bmatrix}.

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D φ = [ φ 1 x 1 φ 1 x 2 φ 1 x n φ 2 x 1 φ 2 x 2 φ 2 x n φ m x 1 φ m x 2 φ m x n ] . D φ superscript φ 1 superscript x 1 superscript φ 1 superscript x 2 normal-… superscript φ 1 superscript x n superscript φ 2 superscript x 1 superscript φ 2 superscript x 2 normal-… superscript φ 2 superscript x n normal-⋮ normal-⋮ normal-⋱ normal-⋮ superscript φ m superscript x 1 superscript φ m superscript x 2 normal-… superscript φ m superscript x n D\varphi=\begin{bmatrix}\frac{\partial\varphi^{1}}{\partial x^{1}}&\frac{% \partial\varphi^{1}}{\partial x^{2}}&\dots&\frac{\partial\varphi^{1}}{\partial x% ^{n}}\\ \frac{\partial\varphi^{2}}{\partial x^{1}}&\frac{\partial\varphi^{2}}{\partial x% ^{2}}&\dots&\frac{\partial\varphi^{2}}{\partial x^{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\varphi^{m}}{\partial x^{1}}&\frac{\partial\varphi^{m}}{\partial x% ^{2}}&\dots&\frac{\partial\varphi^{m}}{\partial x^{n}}\\ \end{bmatrix}.

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= - ( 2 θ 1 2 2 θ 1 θ 2 2 θ 1 θ n 2 θ 2 θ 1 2 θ 2 2 2 θ 2 θ n 2 θ n θ 1 2 θ n θ 2 2 θ n 2 ) ( θ ) | θ = θ * absent evaluated-at superscript 2 superscript subscript θ 1 2 superscript 2 subscript θ 1 subscript θ 2 normal-⋯ superscript 2 subscript θ 1 subscript θ n superscript 2 subscript θ 2 subscript θ 1 superscript 2 superscript subscript θ 2 2 normal-⋯ superscript 2 subscript θ 2 subscript θ n normal-⋮ normal-⋮ normal-⋱ normal-⋮ superscript 2 subscript θ n subscript θ 1 superscript 2 subscript θ n subscript θ 2 normal-⋯ superscript 2 superscript subscript θ n 2 normal-ℓ θ θ superscript θ =-\left.\left(\begin{array}[]{cccc}\tfrac{\partial^{2}}{\partial\theta_{1}^{2}% }&\tfrac{\partial^{2}}{\partial\theta_{1}\partial\theta_{2}}&\cdots&\tfrac{% \partial^{2}}{\partial\theta_{1}\partial\theta_{n}}\\ \tfrac{\partial^{2}}{\partial\theta_{2}\partial\theta_{1}}&\tfrac{\partial^{2}% }{\partial\theta_{2}^{2}}&\cdots&\tfrac{\partial^{2}}{\partial\theta_{2}% \partial\theta_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \tfrac{\partial^{2}}{\partial\theta_{n}\partial\theta_{1}}&\tfrac{\partial^{2}% }{\partial\theta_{n}\partial\theta_{2}}&\cdots&\tfrac{\partial^{2}}{\partial% \theta_{n}^{2}}\\ \end{array}\right)\ell(\theta)\right|_{\theta=\theta^{*}}

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2 f y x = y ( f x ) = ( f x ) y = f x y = y x f . superscript 2 f y x y f x subscript subscript f x y subscript f x y subscript y x f \frac{\partial^{2}f}{\partial y\,\partial x}=\frac{\partial}{\partial y}\left(% \frac{\partial f}{\partial x}\right)=(f_{x})_{y}=f_{xy}=\partial_{yx}f.

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f x ( 0 , r o ) = 0 , 2 f x 2 ( 0 , r o ) = 0 , 3 f x 3 ( 0 , r o ) 0 , f r ( 0 , r o ) = 0 , 2 f r x ( 0 , r o ) 0. f x 0 subscript r o 0 superscript 2 f superscript x 2 0 subscript r o 0 superscript 3 f superscript x 3 0 subscript r o 0 f r 0 subscript r o 0 superscript 2 f r x 0 subscript r o 0. missing-subexpression \begin{array}[]{lll}\displaystyle\frac{\partial f}{\partial x}(0,r_{o})=0,&% \displaystyle\frac{\partial^{2}f}{\partial x^{2}}(0,r_{o})=0,&\displaystyle% \frac{\partial^{3}f}{\partial x^{3}}(0,r_{o})\neq 0,\\ \displaystyle\frac{\partial f}{\partial r}(0,r_{o})=0,&\displaystyle\frac{% \partial^{2}f}{\partial r\partial x}(0,r_{o})\neq 0.\end{array}

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λ i j = 2 f z i z ¯ j subscript λ i j superscript 2 f subscript z i subscript normal-¯ z j \lambda_{ij}=\frac{\partial^{2}f}{\partial z_{i}\partial\bar{z}_{j}}

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H ( f ) = [ 2 f x 2 2 f x y 2 f x z 2 f y x 2 f y 2 2 f y z 2 f z x 2 f z y 2 f z 2 ] , H f superscript 2 f superscript x 2 superscript 2 f x y superscript 2 f x z absent superscript 2 f y x superscript 2 f superscript y 2 superscript 2 f y z absent superscript 2 f z x superscript 2 f z y superscript 2 f superscript z 2 H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{% \partial x\,\partial y}&\frac{\partial^{2}f}{\partial x\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{% 2}}&\frac{\partial^{2}f}{\partial y\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial z\,\partial x}&\frac{\partial^{2}f}{\partial z\,% \partial y}&\frac{\partial^{2}f}{\partial z^{2}}\end{bmatrix},

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J = [ Δ P θ Δ P | V | Δ Q θ Δ Q | V | ] J normal-Δ P θ normal-Δ P V normal-Δ Q θ normal-Δ Q V J=\begin{bmatrix}\dfrac{\partial\Delta P}{\partial\theta}&\dfrac{\partial% \Delta P}{\partial|V|}\\ \dfrac{\partial\Delta Q}{\partial\theta}&\dfrac{\partial\Delta Q}{\partial|V|}% \end{bmatrix}

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ρ D u i D t = - p x i + μ ( 2 u i x j x j ) , ρ D subscript u i D t p subscript x i μ superscript 2 subscript u i subscript x j subscript x j \rho\frac{Du_{i}}{Dt}=-\frac{\partial p}{\partial x_{i}}+\mu\left(\frac{% \partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}\right),

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W i j k l = 2 F i j x k x l + 2 F k l x i x j - 2 F i l x j x k - 2 F j k x i x l subscript W i j k l superscript 2 subscript F i j subscript x k subscript x l superscript 2 subscript F k l subscript x i subscript x j superscript 2 subscript F i l subscript x j subscript x k superscript 2 subscript F j k subscript x i subscript x l W_{ijkl}=\frac{\partial^{2}F_{ij}}{\partial x_{k}\partial x_{l}}+\frac{% \partial^{2}F_{kl}}{\partial x_{i}\partial x_{j}}-\frac{\partial^{2}F_{il}}{% \partial x_{j}\partial x_{k}}-\frac{\partial^{2}F_{jk}}{\partial x_{i}\partial x% _{l}}

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( 1 c ϕ t ϕ x ϕ y ϕ z ) = ( 1 c ϕ t ϕ x ϕ y ϕ z ) ( γ - β γ 0 0 - β γ γ 0 0 0 0 1 0 0 0 0 1 ) . 1 c ϕ superscript t normal-′ ϕ superscript x normal-′ ϕ superscript y normal-′ ϕ superscript z normal-′ 1 c ϕ t ϕ x ϕ y ϕ z γ β γ 0 0 β γ γ 0 0 0 0 1 0 0 0 0 1 \begin{pmatrix}\frac{1}{c}\frac{\partial\phi}{\partial t^{\prime}}&\frac{% \partial\phi}{\partial x^{\prime}}&\frac{\partial\phi}{\partial y^{\prime}}&% \frac{\partial\phi}{\partial z^{\prime}}\end{pmatrix}=\begin{pmatrix}\frac{1}{% c}\frac{\partial\phi}{\partial t}&\frac{\partial\phi}{\partial x}&\frac{% \partial\phi}{\partial y}&\frac{\partial\phi}{\partial z}\end{pmatrix}\begin{% pmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\,.

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L = i = 1 n b i ( x ) x i + i , j = 1 n a i j ( x ) 2 x i x j , L superscript subscript i 1 n subscript b i x subscript x i superscript subscript i j 1 n subscript a i j x superscript 2 subscript x i subscript x j L=\sum_{i=1}^{n}b_{i}(x)\frac{\partial}{\partial x_{i}}+\sum_{i,j=1}^{n}a_{ij}% (x)\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}},

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2 Π x i x j superscript 2 normal-Π subscript x i subscript x j \frac{\partial^{2}\Pi}{\partial x_{i}\partial x_{j}}

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[ d x 1 * d p 1 d x 2 * d p 1 d y 2 * d p 1 ] = [ 2 U x x 1 x 1 2 U x x 1 x 2 2 U x x 1 y 2 2 U x x 1 x 2 2 U x x 2 x 2 2 U x y 2 x 2 2 U y x 1 y 2 2 U y x 2 y 2 2 U y y 2 y 2 ] - 1 [ - 2 U x p 1 x 1 - 2 U x p 1 x 2 - 2 U y p 1 y 2 ] d superscript subscript x 1 d subscript p 1 d superscript subscript x 2 d subscript p 1 d superscript subscript y 2 d subscript p 1 superscript superscript 2 subscript U x subscript x 1 subscript x 1 superscript 2 subscript U x subscript x 1 subscript x 2 superscript 2 subscript U x subscript x 1 subscript y 2 superscript 2 subscript U x subscript x 1 subscript x 2 superscript 2 subscript U x subscript x 2 subscript x 2 superscript 2 subscript U x subscript y 2 subscript x 2 superscript 2 subscript U y subscript x 1 subscript y 2 superscript 2 subscript U y subscript x 2 subscript y 2 superscript 2 subscript U y subscript y 2 subscript y 2 1 superscript 2 subscript U x subscript p 1 subscript x 1 superscript 2 subscript U x subscript p 1 subscript x 2 superscript 2 subscript U y subscript p 1 subscript y 2 \begin{bmatrix}\dfrac{dx_{1}^{*}}{dp_{1}}\\ \dfrac{dx_{2}^{*}}{dp_{1}}\\ \dfrac{dy_{2}^{*}}{dp_{1}}\end{bmatrix}=\begin{bmatrix}\dfrac{\partial^{2}U_{x% }}{\partial x_{1}\partial x_{1}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}% \partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial y_{2}}\\ \dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial x_{2}}&\dfrac{\partial^{2}U_{% x}}{\partial x_{2}\partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial y_{2}% \partial x_{2}}\\ \dfrac{\partial^{2}U_{y}}{\partial x_{1}\partial y_{2}}&\dfrac{\partial^{2}U_{% y}}{\partial x_{2}\partial y_{2}}&\dfrac{\partial^{2}U_{y}}{\partial y_{2}% \partial y_{2}}\end{bmatrix}^{-1}\begin{bmatrix}-\dfrac{\partial^{2}U_{x}}{% \partial p_{1}\partial x_{1}}\\ -\dfrac{\partial^{2}U_{x}}{\partial p_{1}\partial x_{2}}\\ -\dfrac{\partial^{2}U_{y}}{\partial p_{1}\partial y_{2}}\end{bmatrix}

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σ x = 2 Φ y y z z + 2 Φ z z y y - 2 2 Φ y z y z subscript σ x superscript 2 subscript normal-Φ y y z z superscript 2 subscript normal-Φ z z y y 2 superscript 2 subscript normal-Φ y z y z \sigma_{x}=\frac{\partial^{2}\Phi_{yy}}{\partial z\partial z}+\frac{\partial^{% 2}\Phi_{zz}}{\partial y\partial y}-2\frac{\partial^{2}\Phi_{yz}}{\partial y% \partial z}

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[ | U e 1 | 2 | U e 2 | 2 | U e 3 | 2 | U μ 1 | 2 | U μ 2 | 2 | U μ 3 | 2 | U τ 1 | 2 | U τ 2 | 2 | U τ 3 | 2 ] = [ 2 3 1 3 0 1 6 1 3 1 2 1 6 1 3 1 2 ] . superscript subscript U e 1 2 superscript subscript U e 2 2 superscript subscript U e 3 2 superscript subscript U μ 1 2 superscript subscript U μ 2 2 superscript subscript U μ 3 2 superscript subscript U τ 1 2 superscript subscript U τ 2 2 superscript subscript U τ 3 2 2 3 1 3 0 1 6 1 3 1 2 1 6 1 3 1 2 \begin{bmatrix}|U_{e1}|^{2}&|U_{e2}|^{2}&|U_{e3}|^{2}\\ |U_{\mu 1}|^{2}&|U_{\mu 2}|^{2}&|U_{\mu 3}|^{2}\\ |U_{\tau 1}|^{2}&|U_{\tau 2}|^{2}&|U_{\tau 3}|^{2}\end{bmatrix}=\begin{bmatrix% }\frac{2}{3}&\frac{1}{3}&0\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\end{bmatrix}.

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U = [ 1 3 1 3 1 3 ω 3 1 3 ω ¯ 3 ω ¯ 3 1 3 ω 3 ] ( | U i α | 2 ) = [ 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 ] U 1 3 1 3 1 3 ω 3 1 3 normal-¯ ω 3 normal-¯ ω 3 1 3 ω 3 normal-⇒ superscript subscript U i α 2 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 U=\begin{bmatrix}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\ \frac{\omega}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{\bar{\omega}}{\sqrt{3}}\\ \frac{\bar{\omega}}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{\omega}{\sqrt{3}}\end{% bmatrix}\Rightarrow(|U_{i\alpha}|^{2})=\begin{bmatrix}\frac{1}{3}&\frac{1}{3}&% \frac{1}{3}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}

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S σ 2 = i = 1 m ( X i - X ¯ ) 2 , where X ¯ = S μ m formulae-sequence subscript S superscript σ 2 superscript subscript i 1 m superscript subscript X i normal-¯ X 2 where normal-¯ X subscript S μ m S_{\sigma^{2}}=\sum_{i=1}^{m}(X_{i}-\overline{X})^{2},\,\text{ where }% \overline{X}=\frac{S_{\mu}}{m}

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variance = s 2 = i = 1 n ( x i - x ¯ ) 2 n - 1 variance superscript s 2 superscript subscript i 1 n superscript subscript x i normal-¯ x 2 n 1 \mathrm{variance}=s^{2}=\displaystyle\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{% n-1}\!

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C n = i = 1 n ( x i - x ¯ n ) ( y i - y ¯ n ) subscript C n superscript subscript i 1 n subscript x i subscript normal-¯ x n subscript y i subscript normal-¯ y n \textstyle C_{n}=\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})(y_{i}-\bar{y}_{n})

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r ( α , β ) = i ( j i α - j α ¯ ) ( k i β - k β ¯ ) i ( j i α - j α ¯ ) 2 i ( k i β - k β ¯ ) 2 . r α β subscript i subscript superscript j α i normal-¯ superscript j α subscript superscript k β i normal-¯ superscript k β subscript i superscript subscript superscript j α i normal-¯ superscript j α 2 subscript i superscript subscript superscript k β i normal-¯ superscript k β 2 r(\alpha,\beta)=\frac{\sum_{i}(j^{\alpha}_{i}-\bar{j^{\alpha}})(k^{\beta}_{i}-% \bar{k^{\beta}})}{\sqrt{\sum_{i}(j^{\alpha}_{i}-\bar{j^{\alpha}})^{2}}\sqrt{% \sum_{i}(k^{\beta}_{i}-\bar{k^{\beta}})^{2}}}.

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E ( s 2 ) = E ( i = 1 n ( x i - x ¯ ) 2 n - 1 ) = 1 n - 1 E ( i = 1 n [ x i - μ - ( x ¯ - μ ) ] 2 ) = 1 n - 1 [ i = 1 n Var ( x i ) - n Var ( x ¯ ) ] normal-E superscript s 2 absent normal-E superscript subscript i 1 n superscript subscript x i normal-¯ x 2 n 1 missing-subexpression absent 1 n 1 normal-E superscript subscript i 1 n superscript delimited-[] subscript x i μ normal-¯ x μ 2 missing-subexpression absent 1 n 1 delimited-[] superscript subscript i 1 n Var subscript x i n Var normal-¯ x \begin{aligned}\displaystyle\operatorname{E}(s^{2})&\displaystyle=% \operatorname{E}\left(\sum_{i=1}^{n}\frac{(x_{i}-\overline{x})^{2}}{n-1}\right% )\\ &\displaystyle=\frac{1}{n-1}\operatorname{E}\left(\sum_{i=1}^{n}\left[x_{i}-% \mu-\left(\overline{x}-\mu\right)\right]^{2}\right)\\ &\displaystyle=\frac{1}{n-1}\left[\sum_{i=1}^{n}\operatorname{Var}\left(x_{i}% \right)-n\operatorname{Var}\left(\overline{x}\right)\right]\end{aligned}

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i = 1 n ( X i - X ¯ ) 2 superscript subscript i 1 n superscript subscript X i normal-¯ X 2 \sum_{i=1}^{n}(X_{i}-\overline{X})^{2}

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s x = s y | m | 1 n + 1 k + ( y u n k - y ¯ ) 2 m 2 ( x i - x ¯ ) 2 subscript s x subscript s y m 1 n 1 k superscript subscript y u n k normal-¯ y 2 superscript m 2 superscript subscript x i normal-¯ x 2 s_{x}=\frac{s_{y}}{|m|}\sqrt{\frac{1}{n}+\frac{1}{k}+\frac{(y_{unk}-\bar{y})^{% 2}}{m^{2}\sum{(x_{i}-\bar{x})^{2}}}}

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i = 1 n ( X i - X ¯ ) 2 σ 2 χ n - 1 2 similar-to superscript subscript i 1 n superscript subscript X i normal-¯ X 2 superscript σ 2 subscript superscript χ 2 n 1 \sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\sim\sigma^{2}\chi^{2}_{n-1}

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simil ( x , y ) = i I x y ( r x , i - r x ¯ ) ( r y , i - r y ¯ ) i I x y ( r x , i - r x ¯ ) 2 i I x y ( r y , i - r y ¯ ) 2 simil x y subscript i subscript I x y subscript r x i normal-¯ subscript r x subscript r y i normal-¯ subscript r y subscript i subscript I x y superscript subscript r x i normal-¯ subscript r x 2 subscript i subscript I x y superscript subscript r y i normal-¯ subscript r y 2 \operatorname{simil}(x,y)=\frac{\sum\limits_{i\in I_{xy}}(r_{x,i}-\bar{r_{x}})% (r_{y,i}-\bar{r_{y}})}{\sqrt{\sum\limits_{i\in I_{xy}}(r_{x,i}-\bar{r_{x}})^{2% }\sum\limits_{i\in I_{xy}}(r_{y,i}-\bar{r_{y}})^{2}}}

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s x y = 1 N n = 1 N ( x n - x ¯ ) ( y n - y ¯ ) . subscript s x y 1 N superscript subscript n 1 N subscript x n normal-¯ x subscript y n normal-¯ y s_{xy}=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-\bar{x})(y_{n}-\bar{y}).

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𝐂 = i = 1 n ( 𝐱 𝐢 - 𝐱 ¯ ) ( 𝐱 𝐢 - 𝐱 ¯ ) T 𝐂 superscript subscript i 1 n subscript 𝐱 𝐢 normal-¯ 𝐱 superscript subscript 𝐱 𝐢 normal-¯ 𝐱 T \mathbf{C}=\sum_{i=1}^{n}(\mathbf{x_{i}}-\mathbf{\bar{x}})(\mathbf{x_{i}}-% \mathbf{\bar{x}})^{T}

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r x y = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) ( n - 1 ) s x s y = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 i = 1 n ( y i - y ¯ ) 2 , subscript r x y superscript subscript i 1 n subscript x i normal-¯ x subscript y i normal-¯ y n 1 subscript s x subscript s y superscript subscript i 1 n subscript x i normal-¯ x subscript y i normal-¯ y superscript subscript i 1 n superscript subscript x i normal-¯ x 2 superscript subscript i 1 n superscript subscript y i normal-¯ y 2 r_{xy}=\frac{\sum\limits_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{(n-1)s_{x}s_% {y}}=\frac{\sum\limits_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum% \limits_{i=1}^{n}(x_{i}-\bar{x})^{2}\sum\limits_{i=1}^{n}(y_{i}-\bar{y})^{2}}},

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similarity = cos ( θ ) = A B A B = i = 1 n A i × B i i = 1 n ( A i ) 2 × i = 1 n ( B i ) 2 similarity θ normal-⋅ A B norm A norm B superscript subscript i 1 n subscript A i subscript B i superscript subscript i 1 n superscript subscript A i 2 superscript subscript i 1 n superscript subscript B i 2 \,\text{similarity}=\cos(\theta)={A\cdot B\over\|A\|\|B\|}=\frac{\sum\limits_{% i=1}^{n}{A_{i}\times B_{i}}}{\sqrt{\sum\limits_{i=1}^{n}{(A_{i})^{2}}}\times% \sqrt{\sum\limits_{i=1}^{n}{(B_{i})^{2}}}}

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g 1 = m 3 m 2 3 / 2 = 1 n i = 1 n ( x i - x ¯ ) 3 ( 1 n i = 1 n ( x i - x ¯ ) 2 ) 3 / 2 , subscript g 1 subscript m 3 superscript subscript m 2 3 2 1 n superscript subscript i 1 n superscript subscript x i normal-¯ x 3 superscript 1 n superscript subscript i 1 n superscript subscript x i normal-¯ x 2 3 2 \displaystyle g_{1}=\frac{m_{3}}{m_{2}^{3/2}}=\frac{\frac{1}{n}\sum_{i=1}^{n}% \left(x_{i}-\bar{x}\right)^{3}}{\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-% \bar{x}\right)^{2}\right)^{3/2}}\ ,

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n ( X ¯ - μ 0 ) i = 1 n ( X i - X ¯ ) 2 / ( n - 1 ) n normal-¯ X subscript μ 0 superscript subscript i 1 n superscript subscript X i normal-¯ X 2 n 1 \frac{\sqrt{n}(\bar{X}-\mu_{0})}{\sqrt{\sum\limits_{i=1}^{n}(X_{i}-\bar{X})^{2% }/(n-1)}}

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SSE = i = 1 n ( X i - X ¯ ) 2 + i = 1 n ( Y i - Y ¯ ) 2 + i = 1 n ( Z i - Z ¯ ) 2 SSE superscript subscript i 1 n superscript subscript X i normal-¯ X 2 superscript subscript i 1 n superscript subscript Y i normal-¯ Y 2 superscript subscript i 1 n superscript subscript Z i normal-¯ Z 2 \,\text{SSE}=\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}+\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{% 2}+\sum_{i=1}^{n}(Z_{i}-\bar{Z})^{2}

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k = 1 n ( x k - x ¯ ) ( y k - y ¯ ) k = 1 n ( x k - x ¯ ) 2 k = 1 n ( y k - y ¯ ) 2 superscript subscript k 1 n subscript x k normal-¯ x subscript y k normal-¯ y superscript subscript k 1 n superscript subscript x k normal-¯ x 2 superscript subscript k 1 n superscript subscript y k normal-¯ y 2 \frac{\sum_{k=1}^{n}(x_{k}-\bar{x})(y_{k}-\bar{y})}{\sqrt{\sum_{k=1}^{n}(x_{k}% -\bar{x})^{2}\sum_{k=1}^{n}(y_{k}-\bar{y})^{2}}}

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β ^ = 1 T t = 1 T ( x t - x ¯ ) ( y t - y ¯ ) 1 T t = 1 T ( x t - x ¯ ) 2 , normal-^ β 1 T superscript subscript t 1 T subscript x t normal-¯ x subscript y t normal-¯ y 1 T superscript subscript t 1 T superscript subscript x t normal-¯ x 2 \hat{\beta}=\frac{\tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})(y_{t}-\bar{y})}{% \tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})^{2}}\,,

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( x ¯ , Σ ) det ( Σ ) - n / 2 exp ( - 1 2 i = 1 n ( x i - x ¯ ) T Σ - 1 ( x i - x ¯ ) ) , proportional-to normal-¯ x normal-Σ superscript normal-Σ n 2 1 2 superscript subscript i 1 n superscript subscript x i normal-¯ x normal-T superscript normal-Σ 1 subscript x i normal-¯ x \mathcal{L}(\overline{x},\Sigma)\propto\det(\Sigma)^{-n/2}\exp\left(-{1\over 2% }\sum_{i=1}^{n}(x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1}(x_{i}-\overline{x}% )\right),

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S n = 1 n i = 1 n ( X i - X ¯ ) ( X i - X ¯ ) T S n 1 n superscript subscript i 1 n subscript X i normal-¯ X superscript subscript X i normal-¯ X normal-T {S\over n}={1\over n}\sum_{i=1}^{n}(X_{i}-\overline{X})(X_{i}-\overline{X})^{% \mathrm{T}}

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i = 1 n ( x i - μ ) ( x i - μ ) T = i = 1 n ( x i - x ¯ ) ( x i - x ¯ ) T = S superscript subscript i 1 n subscript x i μ superscript subscript x i μ normal-T superscript subscript i 1 n subscript x i normal-¯ x superscript subscript x i normal-¯ x normal-T S \sum_{i=1}^{n}(x_{i}-\mu)(x_{i}-\mu)^{\mathrm{T}}=\sum_{i=1}^{n}(x_{i}-\bar{x}% )(x_{i}-\bar{x})^{\mathrm{T}}=S

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i = 1 n 2 ( y ^ i - y ¯ ) ( y i - y ^ i ) = i = 1 n 2 b ^ ( ( y i - y ¯ ) ( x i - x ¯ ) - b ^ ( x i - x ¯ ) 2 ) = 2 b ^ ( i = 1 n ( y i - y ¯ ) ( x i - x ¯ ) - b ^ i = 1 n ( x i - x ¯ ) 2 ) = 2 b ^ i = 1 n ( ( y i - y ¯ ) ( x i - x ¯ ) - ( y i - y ¯ ) ( x i - x ¯ ) ) = 2 b ^ 0 = 0. superscript subscript i 1 n 2 subscript normal-^ y i normal-¯ y subscript y i subscript normal-^ y i absent superscript subscript i 1 n 2 normal-^ b subscript y i normal-¯ y subscript x i normal-¯ x normal-^ b superscript subscript x i normal-¯ x 2 missing-subexpression absent 2 normal-^ b superscript subscript i 1 n subscript y i normal-¯ y subscript x i normal-¯ x normal-^ b superscript subscript i 1 n superscript subscript x i normal-¯ x 2 missing-subexpression absent 2 normal-^ b superscript subscript i 1 n subscript y i normal-¯ y subscript x i normal-¯ x subscript y i normal-¯ y subscript x i normal-¯ x missing-subexpression absent normal-⋅ 2 normal-^ b 0 0. \begin{aligned}\displaystyle\sum_{i=1}^{n}2(\hat{y}_{i}-\bar{y})(y_{i}-\hat{y}% _{i})&\displaystyle=\sum_{i=1}^{n}2\hat{b}\left((y_{i}-\bar{y})(x_{i}-\bar{x})% -\hat{b}(x_{i}-\bar{x})^{2}\right)\\ &\displaystyle=2\hat{b}\left(\sum_{i=1}^{n}(y_{i}-\bar{y})(x_{i}-\bar{x})-\hat% {b}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)\\ &\displaystyle=2\hat{b}\sum_{i=1}^{n}\left((y_{i}-\bar{y})(x_{i}-\bar{x})-(y_{% i}-\bar{y})(x_{i}-\bar{x})\right)\\ &\displaystyle=2\hat{b}\cdot 0=0.\end{aligned}

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b ^ = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 , normal-^ b superscript subscript i 1 n subscript x i normal-¯ x subscript y i normal-¯ y superscript subscript i 1 n superscript subscript x i normal-¯ x 2 \hat{b}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i=1}^{n}(x_{% i}-\bar{x})^{2}},

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S X 2 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 and S Y 2 = 1 m - 1 i = 1 m ( Y i - Y ¯ ) 2 superscript subscript S X 2 1 n 1 superscript subscript i 1 n superscript subscript X i normal-¯ X 2 and superscript subscript S Y 2 1 m 1 superscript subscript i 1 m superscript subscript Y i normal-¯ Y 2 S_{X}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}\,% \text{ and }S_{Y}^{2}=\frac{1}{m-1}\sum_{i=1}^{m}\left(Y_{i}-\overline{Y}% \right)^{2}

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r = i = 1 N ( X i - X ¯ ) ( Y i - Y ¯ ) i = 1 N ( X i - X ¯ ) 2 i = 1 N ( Y i - Y ¯ ) 2 r subscript superscript N i 1 subscript X i normal-¯ X subscript Y i normal-¯ Y subscript superscript N i 1 superscript subscript X i normal-¯ X 2 subscript superscript N i 1 superscript subscript Y i normal-¯ Y 2 r=\frac{\sum^{N}_{i=1}(X_{i}-\bar{X})(Y_{i}-\bar{Y})}{\sqrt{\sum^{N}_{i=1}(X_{% i}-\bar{X})^{2}}\sqrt{\sum^{N}_{i=1}(Y_{i}-\bar{Y})^{2}}}

NTCIR12-MathWiki-20rate: 1

S S err = i = 1 N ( y i - y i ^ ) 2 S S tot = i = 1 N ( y i - y ¯ ) 2 S S reg = i = 1 N ( y i ^ - y ¯ ) 2 and y ¯ = 1 N y i i = 1 N . S subscript S err absent superscript subscript i 1 N superscript subscript y i normal-^ subscript y i 2 S subscript S tot absent superscript subscript i 1 N superscript subscript y i normal-¯ y 2 S subscript S reg absent superscript subscript i 1 N superscript normal-^ subscript y i normal-¯ y 2 and normal-¯ y absent 1 N subscript superscript subscript y i N i 1 \begin{aligned}\displaystyle SS_{\rm err}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}% -\widehat{y_{i}})^{2}\\ \displaystyle SS_{\rm tot}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}-\bar{y})^{2}\\ \displaystyle SS_{\rm reg}&\displaystyle=\sum_{i=1}^{N}\;(\widehat{y_{i}}-\bar% {y})^{2}\,\text{ and}\\ \displaystyle\bar{y}&\displaystyle=\frac{1}{N}\sum{}_{i=1}^{N}\;y_{i}.\end{aligned}

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g ( β ) = - ψ ( 1 / β ) β 2 - ψ ( 1 / β ) β 3 + 1 β 2 - i = 1 N | x i - μ | β ( log | x i - μ | ) 2 i = 1 N | x i - μ | β + ( i = 1 N | x i - μ | β log | x i - μ | ) 2 ( i = 1 N | x i - μ | β ) 2 + i = 1 N | x i - μ | β log | x i - μ | β i = 1 N | x i - μ | β - log ( β N i = 1 N | x i - μ | β ) β 2 , superscript g normal-′ β ψ 1 β superscript β 2 superscript ψ normal-′ 1 β superscript β 3 1 superscript β 2 superscript subscript i 1 N superscript subscript x i μ β superscript subscript x i μ 2 superscript subscript i 1 N superscript subscript x i μ β superscript superscript subscript i 1 N superscript subscript x i μ β subscript x i μ 2 superscript superscript subscript i 1 N superscript subscript x i μ β 2 superscript subscript i 1 N superscript subscript x i μ β subscript x i μ β superscript subscript i 1 N superscript subscript x i μ β β N superscript subscript i 1 N superscript subscript x i μ β superscript β 2 g^{\prime}(\beta)=-\frac{\psi(1/\beta)}{\beta^{2}}-\frac{\psi^{\prime}(1/\beta% )}{\beta^{3}}+\frac{1}{\beta^{2}}-\frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}(\log% |x_{i}-\mu|)^{2}}{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}}+\frac{(\sum_{i=1}^{N}|x_{% i}-\mu|^{\beta}\log|x_{i}-\mu|)^{2}}{(\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})^{2}}+% \frac{\sum_{i=1}^{N}|x_{i}-\mu|^{\beta}\log|x_{i}-\mu|}{\beta\sum_{i=1}^{N}|x_% {i}-\mu|^{\beta}}-\frac{\log(\frac{\beta}{N}\sum_{i=1}^{N}|x_{i}-\mu|^{\beta})% }{\beta^{2}},

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q x y = ( x - x ¯ ) ( y - y ¯ ) w ( x - x ¯ , y - y ¯ ) I ( x , y ) w ( x - x ¯ , y - y ¯ ) I ( x , y ) subscript q x y x normal-¯ x y normal-¯ y w x normal-¯ x y normal-¯ y I x y w x normal-¯ x y normal-¯ y I x y q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})w(x-\bar{x},y-\bar{y})I(x,y)}{\sum w(x-% \bar{x},y-\bar{y})I(x,y)}

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q x y = ( x - x ¯ ) ( y - y ¯ ) I ( x , y ) I ( x , y ) subscript q x y x normal-¯ x y normal-¯ y I x y I x y q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})I(x,y)}{\sum I(x,y)}

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v = 1 k i = 1 k ( m i - m ¯ ) 2 v 1 k superscript subscript i 1 k superscript subscript m i normal-¯ m 2 \textstyle v=\frac{1}{k}\sum_{i=1}^{k}(m_{i}-\bar{m})^{2}

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𝐖 = 1 n - 1 i = 1 n ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) 𝐖 1 n 1 superscript subscript i 1 n subscript 𝐱 i normal-¯ 𝐱 superscript subscript 𝐱 i normal-¯ 𝐱 normal-′ {\mathbf{W}}=\frac{1}{n-1}\sum_{i=1}^{n}(\mathbf{x}_{i}-\overline{\mathbf{x}})% (\mathbf{x}_{i}-\overline{\mathbf{x}})^{\prime}

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𝐖 = i = 1 n x ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) + i = 1 n y ( 𝐲 i - 𝐲 ¯ ) ( 𝐲 i - 𝐲 ¯ ) n x + n y - 2 𝐖 superscript subscript i 1 subscript n x subscript 𝐱 i normal-¯ 𝐱 superscript subscript 𝐱 i normal-¯ 𝐱 normal-′ superscript subscript i 1 subscript n y subscript 𝐲 i normal-¯ 𝐲 superscript subscript 𝐲 i normal-¯ 𝐲 normal-′ subscript n x subscript n y 2 {\mathbf{W}}=\frac{\sum_{i=1}^{n_{x}}(\mathbf{x}_{i}-\overline{\mathbf{x}})(% \mathbf{x}_{i}-\overline{\mathbf{x}})^{\prime}+\sum_{i=1}^{n_{y}}(\mathbf{y}_{% i}-\overline{\mathbf{y}})(\mathbf{y}_{i}-\overline{\mathbf{y}})^{\prime}}{n_{x% }+n_{y}-2}

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C = μ ^ 4 σ ^ 4 = 1 n i = 1 n ( x i - x ¯ ) 4 ( 1 n i = 1 n ( x i - x ¯ ) 2 ) 2 , C subscript normal-^ μ 4 superscript normal-^ σ 4 1 n superscript subscript i 1 n superscript subscript x i normal-¯ x 4 superscript 1 n superscript subscript i 1 n superscript subscript x i normal-¯ x 2 2 C=\frac{\hat{\mu}_{4}}{\hat{\sigma}^{4}}=\frac{\frac{1}{n}\sum_{i=1}^{n}(x_{i}% -\bar{x})^{4}}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}},

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S = i = 1 n ( R i - R ¯ ) 2 , S superscript subscript i 1 n superscript subscript R i normal-¯ R 2 S=\sum_{i=1}^{n}(R_{i}-\bar{R})^{2},

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σ X 2 = i = 1 n ( X i - X ¯ ) 2 n . subscript superscript σ 2 X superscript subscript i 1 n superscript subscript X i normal-¯ X 2 n \sigma^{2}_{X}=\frac{\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\,{}}{n}.

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g 2 = m 4 m 2 2 - 3 = 1 n i = 1 n ( x i - x ¯ ) 4 ( 1 n i = 1 n ( x i - x ¯ ) 2 ) 2 - 3 subscript g 2 subscript m 4 superscript subscript m 2 2 3 1 n superscript subscript i 1 n superscript subscript x i normal-¯ x 4 superscript 1 n superscript subscript i 1 n superscript subscript x i normal-¯ x 2 2 3 g_{2}=\frac{m_{4}}{m_{2}^{2}}-3=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-% \overline{x})^{4}}{\left(\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}% \right)^{2}}-3

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= ( n + 1 ) n ( n - 1 ) ( n - 2 ) ( n - 3 ) i = 1 n ( x i - x ¯ ) 4 ( i = 1 n ( x i - x ¯ ) 2 ) 2 - 3 ( n - 1 ) 2 ( n - 2 ) ( n - 3 ) absent n 1 n n 1 n 2 n 3 superscript subscript i 1 n superscript subscript x i normal-¯ x 4 superscript superscript subscript i 1 n superscript subscript x i normal-¯ x 2 2 3 superscript n 1 2 n 2 n 3 =\frac{(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}\;\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{4}% }{\left(\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}}-3\,\frac{(n-1)^{2}}{(n-2% )\,(n-3)}

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i = 1 n ( X i - X ¯ ) 2 σ 2 χ n - 1 2 , similar-to superscript subscript i 1 n superscript subscript X i normal-¯ X 2 superscript σ 2 subscript superscript χ 2 n 1 \sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\sim\sigma^{2}\chi^{2}_{n-1},\quad

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R M D ( S ) = i = 1 n j = 1 n | y i - y j | ( n - 1 ) i = 1 n y i R M D S superscript subscript i 1 n superscript subscript j 1 n subscript y i subscript y j n 1 superscript subscript i 1 n subscript y i RMD(S)=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}|y_{i}-y_{j}|}{(n-1)\sum_{i=1}^{n}y_{% i}}

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σ 2 = i = 1 N w i ( x i - x ¯ * ) 2 i = 1 N w i superscript σ 2 superscript subscript i 1 N subscript w i superscript subscript x i superscript normal-¯ x 2 superscript subscript i 1 N subscript w i \sigma^{2}=\frac{\sum_{i=1}^{N}w_{i}(x_{i}-\overline{x}^{\,*})^{2}}{\sum_{i=1}% ^{N}w_{i}}

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I = N i j w i j i j w i j ( X i - X ¯ ) ( X j - X ¯ ) i ( X i - X ¯ ) 2 I N subscript i subscript j subscript w i j subscript i subscript j subscript w i j subscript X i normal-¯ X subscript X j normal-¯ X subscript i superscript subscript X i normal-¯ X 2 I=\frac{N}{\sum_{i}\sum_{j}w_{ij}}\frac{\sum_{i}\sum_{j}w_{ij}(X_{i}-\bar{X})(% X_{j}-\bar{X})}{\sum_{i}(X_{i}-\bar{X})^{2}}

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s y m b o l Σ ^ = 1 n j = 1 n ( 𝐱 j - 𝐱 ¯ ) ( 𝐱 j - 𝐱 ¯ ) T normal-^ s y m b o l normal-Σ 1 n superscript subscript j 1 n subscript 𝐱 j normal-¯ 𝐱 superscript subscript 𝐱 j normal-¯ 𝐱 T \displaystyle\widehat{symbol\Sigma}={1\over n}\sum_{j=1}^{n}\left(\mathbf{x}_{% j}-\bar{\mathbf{x}}\right)\left(\mathbf{x}_{j}-\bar{\mathbf{x}}\right)^{T}

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F = i = 1 n ( y i ^ - y ¯ ) 2 / k j = 1 k i = 1 n j ( y i j - y i ^ ) 2 / ( n - k - 1 ) F superscript subscript i 1 n superscript normal-^ subscript y i normal-¯ y 2 k superscript subscript j 1 k superscript subscript i 1 subscript n j superscript subscript y i j normal-^ subscript y i 2 n k 1 F=\frac{{\displaystyle\sum_{i=1}^{n}\left(\widehat{y_{i}}-\bar{y}\right)^{2}}/% {k}}{{\displaystyle{\sum_{j=1}^{k}}{\sum_{i=1}^{n_{j}}}\left(y_{ij}-\widehat{y% _{i}}\right)^{2}}/{(n-k-1)}}

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i = 1 n ( y i - y ¯ ) 2 = i = 1 n ( y i - y ¯ + y ^ i - y ^ i ) 2 = i = 1 n ( ( y ^ i - y ¯ ) + ( y i - y ^ i ) ε ^ i ) 2 = i = 1 n ( ( y ^ i - y ¯ ) 2 + 2 ε ^ i ( y ^ i - y ¯ ) + ε ^ i 2 ) = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 + 2 i = 1 n ε ^ i ( y ^ i - y ¯ ) = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 + 2 i = 1 n ε ^ i ( β ^ 0 + β ^ 1 x i 1 + + β ^ p x i p - y ¯ ) = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 + 2 ( β ^ 0 - y ¯ ) i = 1 n ε ^ i 0 + 2 β ^ 1 i = 1 n ε ^ i x i 1 0 + + 2 β ^ p i = 1 n ε ^ i x i p 0 = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 = ESS + RSS superscript subscript i 1 n superscript subscript y i normal-¯ y 2 absent superscript subscript i 1 n superscript subscript y i normal-¯ y subscript normal-^ y i subscript normal-^ y i 2 superscript subscript i 1 n superscript subscript normal-^ y i normal-¯ y subscript normal-⏟ subscript y i subscript normal-^ y i subscript normal-^ ε i 2 missing-subexpression absent superscript subscript i 1 n superscript subscript normal-^ y i normal-¯ y 2 2 subscript normal-^ ε i subscript normal-^ y i normal-¯ y superscript subscript normal-^ ε i 2 missing-subexpression absent superscript subscript i 1 n superscript subscript normal-^ y i normal-¯ y 2 superscript subscript i 1 n superscript subscript normal-^ ε i 2 2 superscript subscript i 1 n subscript normal-^ ε i subscript normal-^ y i normal-¯ y missing-subexpression absent superscript subscript i 1 n superscript subscript normal-^ y i normal-¯ y 2 superscript subscript i 1 n superscript subscript normal-^ ε i 2 2 superscript subscript i 1 n subscript normal-^ ε i subscript normal-^ β 0 subscript normal-^ β 1 subscript x i 1 normal-⋯ subscript normal-^ β p subscript x i p normal-¯ y missing-subexpression absent superscript subscript i 1 n superscript subscript normal-^ y i normal-¯ y 2 superscript subscript i 1 n superscript subscript normal-^ ε i 2 2 subscript normal-^ β 0 normal-¯ y subscript normal-⏟ superscript subscript i 1 n subscript normal-^ ε i 0 2 subscript normal-^ β 1 subscript normal-⏟ superscript subscript i 1 n subscript normal-^ ε i subscript x i 1 0 normal-⋯ 2 subscript normal-^ β p subscript normal-⏟ superscript subscript i 1 n subscript normal-^ ε i subscript x i p 0 missing-subexpression absent superscript subscript i 1 n superscript subscript normal-^ y i normal-¯ y 2 superscript subscript i 1 n superscript subscript normal-^ ε i 2 ESS RSS \begin{aligned}\displaystyle\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}&% \displaystyle=\sum_{i=1}^{n}(y_{i}-\overline{y}+\hat{y}_{i}-\hat{y}_{i})^{2}=% \sum_{i=1}^{n}((\hat{y}_{i}-\bar{y})+\underbrace{(y_{i}-\hat{y}_{i})}_{\hat{% \varepsilon}_{i}})^{2}\\ &\displaystyle=\sum_{i=1}^{n}((\hat{y}_{i}-\bar{y})^{2}+2\hat{\varepsilon}_{i}% (\hat{y}_{i}-\bar{y})+\hat{\varepsilon}_{i}^{2})\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2\sum_{i=1}^{n}\hat{\varepsilon}_{i}(\hat{y}_{i}-\bar{y})% \\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2\sum_{i=1}^{n}\hat{\varepsilon}_{i}(\hat{\beta}_{0}+\hat% {\beta}_{1}x_{i1}+\cdots+\hat{\beta}_{p}x_{ip}-\overline{y})\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2(\hat{\beta}_{0}-\overline{y})\underbrace{\sum_{i=1}^{n}% \hat{\varepsilon}_{i}}_{0}+2\hat{\beta}_{1}\underbrace{\sum_{i=1}^{n}\hat{% \varepsilon}_{i}x_{i1}}_{0}+\cdots+2\hat{\beta}_{p}\underbrace{\sum_{i=1}^{n}% \hat{\varepsilon}_{i}x_{ip}}_{0}\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}=\mathrm{ESS}+\mathrm{RSS}\\ \end{aligned}

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r = r x y = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 i = 1 n ( y i - y ¯ ) 2 r subscript r x y subscript superscript n i 1 subscript x i normal-¯ x subscript y i normal-¯ y subscript superscript n i 1 superscript subscript x i normal-¯ x 2 subscript superscript n i 1 superscript subscript y i normal-¯ y 2 r=r_{xy}=\frac{\sum^{n}_{i=1}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum^{n}_{i% =1}(x_{i}-\bar{x})^{2}}\sqrt{\sum^{n}_{i=1}(y_{i}-\bar{y})^{2}}}

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r = 1 n - 1 i = 1 n ( X i - X ¯ ) ( Y i - Y ¯ ) s X s Y r 1 n 1 superscript subscript i 1 n subscript X i normal-¯ X subscript Y i normal-¯ Y subscript s X subscript s Y r=\frac{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{s% _{X}s_{Y}}

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s n - 1 = 1 n - 1 i = 1 n ( X i - X ¯ ) 2 . subscript s n 1 1 n 1 superscript subscript i 1 n superscript subscript X i normal-¯ X 2 s_{n-1}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}.

NTCIR12-MathWiki-20rate: 1

V a r ( β 1 ) = σ 2 i = 1 n ( x i - x ¯ ) 2 . V a r subscript β 1 absent superscript σ 2 superscript subscript i 1 n superscript subscript x i normal-¯ x 2 \begin{aligned}\displaystyle Var(\beta_{1})&\displaystyle=\frac{\sigma^{2}}{% \sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}}.\end{aligned}

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r = y ¯ x ¯ = i = 1 n y i = 1 n x r normal-¯ y normal-¯ x superscript subscript i 1 n y superscript subscript i 1 n x r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^{n}y}{\sum_{i=1}^{n}x}

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β 1 ^ = ( x i - x ¯ ) ( y i - y ¯ ) ( x i - x ¯ ) 2 and β 0 ^ = y ¯ - β 1 ^ x ¯ normal-^ subscript β 1 subscript x i normal-¯ x subscript y i normal-¯ y superscript subscript x i normal-¯ x 2 and normal-^ subscript β 0 normal-¯ y normal-^ subscript β 1 normal-¯ x \widehat{\beta_{1}}=\frac{\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum(x_{i}-\bar{% x})^{2}}\,\text{ and }\hat{\beta_{0}}=\bar{y}-\widehat{\beta_{1}}\bar{x}

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β ^ = i = 1 n ( x i - x ¯ ) ( y i - y ¯ ) i = 1 n ( x i - x ¯ ) 2 = x y ¯ - x ¯ y ¯ x 2 ¯ - x ¯ 2 = Cov [ x , y ] Var [ x ] = r x y s y s x , normal-^ β superscript subscript i 1 n subscript x i normal-¯ x subscript y i normal-¯ y superscript subscript i 1 n superscript subscript x i normal-¯ x 2 normal-¯ x y normal-¯ x normal-¯ y normal-¯ superscript x 2 superscript normal-¯ x 2 Cov x y Var x subscript r x y subscript s y subscript s x \displaystyle\hat{\beta}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{% \sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}=\frac{\overline{xy}-\bar{x}\bar{y}}{% \overline{x^{2}}-\bar{x}^{2}}=\frac{\operatorname{Cov}[x,y]}{\operatorname{Var% }[x]}=r_{xy}\frac{s_{y}}{s_{x}},

NTCIR12-MathWiki-20rate: 0

𝐐 = 1 N - 1 i = 1 N ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) T , 𝐐 1 N 1 superscript subscript i 1 N subscript 𝐱 i normal-¯ 𝐱 superscript subscript 𝐱 i normal-¯ 𝐱 normal-T \mathbf{Q}={1\over{N-1}}\sum_{i=1}^{N}(\mathbf{x}_{i}-\mathbf{\bar{x}})(% \mathbf{x}_{i}-\mathbf{\bar{x}})^{\mathrm{T}},

NTCIR12-MathWiki-20rate: 1

q j k = i = 1 N w i ( i = 1 N w i ) 2 - i = 1 N w i 2 i = 1 N w i ( x i j - x ¯ j ) ( x i k - x ¯ k ) . subscript q j k superscript subscript i 1 N subscript w i superscript superscript subscript i 1 N subscript w i 2 superscript subscript i 1 N superscript subscript w i 2 superscript subscript i 1 N subscript w i subscript x i j subscript normal-¯ x j subscript x i k subscript normal-¯ x k q_{jk}=\frac{\sum_{i=1}^{N}w_{i}}{\left(\sum_{i=1}^{N}w_{i}\right)^{2}-\sum_{i% =1}^{N}w_{i}^{2}}\sum_{i=1}^{N}w_{i}\left(x_{ij}-\bar{x}_{j}\right)\left(x_{ik% }-\bar{x}_{k}\right).

NTCIR12-MathWiki-20rate: 0

S a = E [ R a - R b ] σ a = E [ R a - R b ] var [ R a - R b ] , subscript S a E delimited-[] subscript R a subscript R b subscript σ a E delimited-[] subscript R a subscript R b var delimited-[] subscript R a subscript R b S_{a}=\frac{E[R_{a}-R_{b}]}{\sigma_{a}}=\frac{E[R_{a}-R_{b}]}{\sqrt{\mathrm{% var}[R_{a}-R_{b}]}},

NTCIR12-MathWiki-20rate: 1

SSE ( α ^ , β ^ ) β ^ = - 2 i = 1 n [ ( y i - y ¯ ) - β ^ ( x i - x ¯ ) ] ( x i - x ¯ ) = 0 SSE normal-^ α normal-^ β normal-^ β 2 superscript subscript i 1 n delimited-[] subscript y i normal-¯ y normal-^ β subscript x i normal-¯ x subscript x i normal-¯ x 0 \displaystyle\frac{\partial\,\mathrm{SSE}\left(\hat{\alpha},\hat{\beta}\right)% }{\partial\hat{\beta}}=-2\sum_{i=1}^{n}\left[\left(y_{i}-\bar{y}\right)-\hat{% \beta}\left(x_{i}-\bar{x}\right)\right]\left(x_{i}-\bar{x}\right)=0

NTCIR12-MathWiki-20rate: 1

β ^ = i = 1 n ( y i - y ¯ ) ( x i - x ¯ ) i = 1 n ( x i - x ¯ ) 2 = C o v ( x , y ) V a r ( x ) normal-^ β absent absent superscript subscript i 1 n subscript y i normal-¯ y subscript x i normal-¯ x superscript subscript i 1 n superscript subscript x i normal-¯ x 2 C o v x y V a r x \displaystyle\hat{\beta}=\frac{}{}\frac{\sum_{i=1}^{n}\left(y_{i}-\bar{y}% \right)\left(x_{i}-\bar{x}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2% }}=\frac{Cov\left(x,y\right)}{Var\left(x\right)}

NTCIR12-MathWiki-20rate: 2

b 1 = m 3 s 3 = 1 n i = 1 n ( x i - x ¯ ) 3 [ 1 n - 1 i = 1 n ( x i - x ¯ ) 2 ] 3 / 2 , subscript b 1 subscript m 3 superscript s 3 1 n superscript subscript i 1 n superscript subscript x i normal-¯ x 3 superscript delimited-[] 1 n 1 superscript subscript i 1 n superscript subscript x i normal-¯ x 2 3 2 b_{1}=\frac{m_{3}}{s^{3}}=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})% ^{3}}{\left[\tfrac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\right]^{3/2}}\ ,

NTCIR12-MathWiki-20rate: 1

λ = E ( V ) stdev ( V ) = i = 1 t c i μ i Var ( i = 1 t c i G i ) = i = 1 t c i μ i i = 1 t c i 2 σ i 2 + 2 i = 1 t j = i c i c j σ i j λ normal-E V stdev V superscript subscript i 1 t subscript c i subscript μ i Var superscript subscript i 1 t subscript c i subscript G i superscript subscript i 1 t subscript c i subscript μ i superscript subscript i 1 t superscript subscript c i 2 superscript subscript σ i 2 2 superscript subscript i 1 t subscript j i subscript c i subscript c j subscript σ i j \lambda=\frac{\operatorname{E}(V)}{\operatorname{stdev}(V)}=\frac{\sum_{i=1}^{% t}c_{i}\mu_{i}}{\sqrt{\,\text{Var}(\sum_{i=1}^{t}c_{i}G_{i})}}=\frac{\sum_{i=1% }^{t}c_{i}\mu_{i}}{\sqrt{\sum_{i=1}^{t}c_{i}^{2}\sigma_{i}^{2}+2\sum_{i=1}^{t}% \sum_{j=i}c_{i}c_{j}\sigma_{ij}}}

NTCIR12-MathWiki-20rate: 1

σ ^ = 1 N - 1.5 i = 1 n ( x i - x ¯ ) 2 , normal-^ σ 1 N 1.5 superscript subscript i 1 n superscript subscript x i normal-¯ x 2 \hat{\sigma}=\sqrt{\frac{1}{N-1.5}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}},

NTCIR12-MathWiki-20rate: 1

S E β ^ = 1 n - 2 i = 1 n ( y i - y ^ i ) 2 i = 1 n ( x i - x ¯ ) 2 S subscript E normal-^ β 1 n 2 superscript subscript i 1 n superscript subscript y i subscript normal-^ y i 2 superscript subscript i 1 n superscript subscript x i normal-¯ x 2 SE_{\widehat{\beta}}=\frac{\sqrt{\frac{1}{n-2}\sum_{i=1}^{n}(y_{i}-\widehat{y}% _{i})^{2}}}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}}

NTCIR12-MathWiki-20rate: 1

var ( ε ^ i ) = σ 2 ( 1 - 1 n - ( x i - x ¯ ) 2 j = 1 n ( x j - x ¯ ) 2 ) . var subscript normal-^ ε i superscript σ 2 1 1 n superscript subscript x i normal-¯ x 2 superscript subscript j 1 n superscript subscript x j normal-¯ x 2 \operatorname{var}(\widehat{\varepsilon}_{i})=\sigma^{2}\left(1-\frac{1}{n}-% \frac{(x_{i}-\bar{x})^{2}}{\sum_{j=1}^{n}(x_{j}-\bar{x})^{2}}\right).

NTCIR12-MathWiki-20rate: 1

f X 1 n ( x 1 n ) = i = 1 n 1 2 π σ 2 e - ( x i - θ ) 2 / ( 2 σ 2 ) = ( 2 π σ 2 ) - n / 2 e - i = 1 n ( x i - θ ) 2 / ( 2 σ 2 ) = ( 2 π σ 2 ) - n / 2 e - i = 1 n ( ( x i - x ¯ ) - ( θ - x ¯ ) ) 2 / ( 2 σ 2 ) = ( 2 π σ 2 ) - n / 2 exp ( - 1 2 σ 2 ( i = 1 n ( x i - x ¯ ) 2 + i = 1 n ( θ - x ¯ ) 2 - 2 i = 1 n ( x i - x ¯ ) ( θ - x ¯ ) ) ) . subscript f superscript subscript X 1 n superscript subscript x 1 n absent superscript subscript product i 1 n 1 2 π superscript σ 2 superscript e superscript subscript x i θ 2 2 superscript σ 2 superscript 2 π superscript σ 2 n 2 superscript e superscript subscript i 1 n superscript subscript x i θ 2 2 superscript σ 2 missing-subexpression absent superscript 2 π superscript σ 2 n 2 superscript e superscript subscript i 1 n superscript subscript x i normal-¯ x θ normal-¯ x 2 2 superscript σ 2 missing-subexpression absent superscript 2 π superscript σ 2 n 2 1 2 superscript σ 2 superscript subscript i 1 n superscript subscript x i normal-¯ x 2 superscript subscript i 1 n superscript θ normal-¯ x 2 2 superscript subscript i 1 n subscript x i normal-¯ x θ normal-¯ x \begin{aligned}\displaystyle f_{X_{1}^{n}}(x_{1}^{n})&\displaystyle=\prod_{i=1% }^{n}\tfrac{1}{\sqrt{2\pi\sigma^{2}}}\,e^{-(x_{i}-\theta)^{2}/(2\sigma^{2})}=(% 2\pi\sigma^{2})^{-n/2}\,e^{-\sum_{i=1}^{n}(x_{i}-\theta)^{2}/(2\sigma^{2})}\\ &\displaystyle=(2\pi\sigma^{2})^{-n/2}\,e^{-\sum_{i=1}^{n}((x_{i}-\overline{x}% )-(\theta-\overline{x}))^{2}/(2\sigma^{2})}\\ &\displaystyle=(2\pi\sigma^{2})^{-n/2}\,\exp\left({-1\over 2\sigma^{2}}\left(% \sum_{i=1}^{n}(x_{i}-\overline{x})^{2}+\sum_{i=1}^{n}(\theta-\overline{x})^{2}% -2\sum_{i=1}^{n}(x_{i}-\overline{x})(\theta-\overline{x})\right)\right).\end{aligned}

NTCIR12-MathWiki-20rate: 1

s = 1 n - 1 i = 1 n ( x i - x ¯ ) 2 , s 1 n 1 superscript subscript i 1 n superscript subscript x i normal-¯ x 2 s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\,,

NTCIR12-MathWiki-20rate: 3

sim ( d j , q ) = 𝐝 𝐣 𝐪 𝐝 𝐣 𝐪 = i = 1 N w i , j w i , q i = 1 N w i , j 2 i = 1 N w i , q 2 sim subscript d j q normal-⋅ subscript 𝐝 𝐣 𝐪 norm subscript 𝐝 𝐣 norm 𝐪 superscript subscript i 1 N subscript w i j subscript w i q superscript subscript i 1 N superscript subscript w i j 2 superscript subscript i 1 N superscript subscript w i q 2 \mathrm{sim}(d_{j},q)=\frac{\mathbf{d_{j}}\cdot\mathbf{q}}{\left\|\mathbf{d_{j% }}\right\|\left\|\mathbf{q}\right\|}=\frac{\sum_{i=1}^{N}w_{i,j}w_{i,q}}{\sqrt% {\sum_{i=1}^{N}w_{i,j}^{2}}\sqrt{\sum_{i=1}^{N}w_{i,q}^{2}}}

NTCIR12-MathWiki-20rate: 3

N C C ( μ , , ) = x i Ω F ( ( x i ) - ¯ ) ( ( T μ ( x i ) ) - ¯ ) x i Ω F ( ( x i ) - ¯ ) 2 x i Ω F ( ( T μ ( x i ) ) - ¯ ) 2 N C C μ subscript subscript subscript subscript x i subscript normal-Ω F subscript subscript x i normal-¯ subscript subscript subscript T μ subscript x i normal-¯ subscript subscript subscript x i subscript normal-Ω F superscript subscript subscript x i normal-¯ subscript 2 subscript subscript x i subscript normal-Ω F superscript subscript subscript T μ subscript x i normal-¯ subscript 2 NCC(\mu,\mathcal{I_{F}},\mathcal{I_{M}})=\dfrac{\sum_{x_{i}\in\Omega_{F}}\left% (\mathcal{I_{F}}(x_{i})-\overline{\mathcal{I_{F}}}\right)\left(\mathcal{I_{M}}% ({T}_{\mu}(x_{i}))-\overline{\mathcal{I_{M}}}\right)}{\sqrt{\sum_{x_{i}\in% \Omega_{F}}\left(\mathcal{I_{F}}(x_{i})-\overline{\mathcal{I_{F}}}\right)^{2}% \sum_{x_{i}\in\Omega_{F}}\left(\mathcal{I_{M}}({T}_{\mu}(x_{i}))-\overline{% \mathcal{I_{M}}}\right)^{2}}}

NTCIR12-MathWiki-3rate: 2

m M o o n = 0.25 + 2.5 log 10 ( 3 2 0.00257 2 ) = - 12.26 subscript m M o o n 0.25 2.5 subscript 10 3 2 superscript 0.00257 2 12.26 m_{Moon}=0.25+2.5\log_{10}{\left(\frac{3}{2}0.00257^{2}\right)}=-12.26\!\,

NTCIR12-MathWiki-3rate: 3

m AB = - 5 2 log 10 ( f ν Jy ) + 8.90. m AB 5 2 subscript 10 subscript f ν Jy 8.90. m\text{AB}=-\frac{5}{2}\log_{10}\left(\frac{f_{\nu}}{\,\text{Jy}}\right)+8.90.

NTCIR12-MathWiki-3rate: 0

C M = - 0.5 π ( A 0 + A 1 - A 2 / 2 ) subscript C M 0.5 π subscript A 0 subscript A 1 subscript A 2 2 \ C_{M}=-0.5\pi(A_{0}+A_{1}-A_{2}/2)

NTCIR12-MathWiki-3rate: 0

cr ( K n ) = 1 4 n 2 n - 1 2 n - 2 2 n - 3 2 . cr subscript K n 1 4 n 2 n 1 2 n 2 2 n 3 2 \textrm{cr}(K_{n})=\frac{1}{4}\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor% \frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor% \frac{n-3}{2}\right\rfloor.

NTCIR12-MathWiki-3rate: 0

m n = - 2 e V 1 ( t f ) 2 m n 2 e subscript V 1 superscript t f 2 \frac{m}{n}=-2eV_{1}\left(\frac{t}{f}\right)^{2}

NTCIR12-MathWiki-3rate: 3

Attenuation (dB) = 10 × log 10 ( Input intensity (W) Output intensity (W) ) Attenuation (dB) 10 subscript 10 Input intensity (W) Output intensity (W) \,\text{Attenuation (dB)}=10\times\log_{10}\left(\frac{\,\text{Input intensity% (W)}}{\,\text{Output intensity (W)}}\right)

NTCIR12-MathWiki-3rate: 0

log 2 ( 0 ) = - 1 subscript 2 0 1 \lfloor\log_{2}(0)\rfloor=-1

NTCIR12-MathWiki-3rate: 1

log 2 N subscript 2 N \lfloor\log_{2}N\rfloor

NTCIR12-MathWiki-3rate: 0

X = 1 2 [ ( T β ) 0.5 - ( T β ) - 0.5 ] X 1 2 delimited-[] superscript T β 0.5 superscript T β 0.5 X=\frac{1}{2}\left[\left(\frac{T}{\beta}\right)^{0.5}-\left(\frac{T}{\beta}% \right)^{-0.5}\right]

NTCIR12-MathWiki-3rate: 1

x = log 2 x + 1 x subscript 2 x 1 x=\lfloor\log_{2}x\rfloor+1

NTCIR12-MathWiki-3rate: 1

r = f 2 ( θ ) - f 1 ( θ ) r subscript f 2 θ subscript f 1 θ r=f_{2}(\theta)-f_{1}(\theta)

NTCIR12-MathWiki-3rate: 0

𝐏 = g ( 𝐚 ) - g ( 𝐛 ) 𝐏 g 𝐚 g 𝐛 \mathbf{P}=g(\mathbf{a})-g(\mathbf{b})

NTCIR12-MathWiki-3rate: 0

α s = arctan ( b R ) - arctan ( b R + r s ) subscript α s b R b R subscript r s \alpha_{s}=\arctan\left(\frac{b}{R}\right)-\arctan\left(\frac{b}{R+r_{s}}\right)

NTCIR12-MathWiki-3rate: 1

k = log 2 n k subscript 2 n k=\lfloor\log_{2}n\rfloor

NTCIR12-MathWiki-3rate: 1

- log ( x ) x -\log(x)

NTCIR12-MathWiki-3rate: 1

- log ( t ) t -\log(t)\,

NTCIR12-MathWiki-3rate: 1

log 2 ( x ) + 2 log 2 ( log 2 ( x ) + 1 ) + 1 subscript 2 x 2 subscript 2 subscript 2 x 1 1 \lfloor\log_{2}(x)\rfloor+2\lfloor\log_{2}(\lfloor\log_{2}(x)\rfloor+1)\rfloor+1

NTCIR12-MathWiki-3rate: 1

2 log 2 ( x ) + 1 2 subscript 2 x 1 2\lfloor\log_{2}(x)\rfloor+1

NTCIR12-MathWiki-3rate: 0

d = - m 2 d m 2 d=-\left\lfloor\frac{m}{2}\right\rfloor

NTCIR12-MathWiki-3rate: 1

γ = 1 - k = 2 ( - 1 ) k log 2 k k + 1 . γ 1 superscript subscript k 2 superscript 1 k subscript 2 k k 1 \gamma=1-\sum_{k=2}^{\infty}(-1)^{k}\frac{\lfloor\log_{2}k\rfloor}{k+1}.

NTCIR12-MathWiki-3rate: 1

log 2 n subscript 2 n \lfloor\log_{2}n\rfloor

NTCIR12-MathWiki-3rate: 0

206.835 - 1.015 ( total words total sentences ) - 84.6 ( total syllables total words ) . 206.835 1.015 total words total sentences 84.6 total syllables total words 206.835-1.015\left(\frac{\,\text{total words}}{\,\text{total sentences}}\right% )-84.6\left(\frac{\,\text{total syllables}}{\,\text{total words}}\right).

NTCIR12-MathWiki-3rate: 0

m n + 2 m n + + ( n - 1 ) m n = n m + 2 n m + + ( m - 1 ) n m . m n 2 m n normal-… n 1 m n n m 2 n m normal-… m 1 n m \left\lfloor\frac{m}{n}\right\rfloor+\left\lfloor\frac{2m}{n}\right\rfloor+% \dots+\left\lfloor\frac{(n-1)m}{n}\right\rfloor=\left\lfloor\frac{n}{m}\right% \rfloor+\left\lfloor\frac{2n}{m}\right\rfloor+\dots+\left\lfloor\frac{(m-1)n}{% m}\right\rfloor.

NTCIR12-MathWiki-3rate: 1

W = ϕ ( b ) - ϕ ( a ) W ϕ b ϕ a W=\phi(b)-\phi(a)

NTCIR12-MathWiki-3rate: 1

log ( x ) = - log ( 1 x ) x 1 x \log\left(x\right)=-\log\left(\frac{1}{x}\right)

NTCIR12-MathWiki-3rate: 1

log ( x ) = - log ( 1 x ) x 1 x \log\left(x\right)=-\log\left(\frac{1}{x}\right)

NTCIR12-MathWiki-3rate: 1

Δ n ( s ) = { x n s } - log 2 ( 1 + s ) fragments subscript normal-Δ n fragments normal-( s normal-) P fragments normal-{ subscript x n s normal-} subscript 2 fragments normal-( 1 s normal-) \Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s)

NTCIR12-MathWiki-3rate: 0

n / 2 - 1 = m n 2 1 m \lfloor n/2-1\rfloor=m

NTCIR12-MathWiki-3rate: 1

M = - 1 log 2 ( 1 - p ) M 1 subscript 2 1 p M=\left\lfloor\frac{-1}{\log_{2}(1-p)}\right\rfloor

NTCIR12-MathWiki-3rate: 0

μ a = μ 0 ( 1 - 0.5 k H ) - 2 subscript μ a subscript μ 0 superscript 1 0.5 k H 2 \mu_{a}={{\mu_{0}}{{(1-0.5kH)}^{-2}}}

NTCIR12-MathWiki-3rate: 2

pH = p K a + log 10 ( [ A - ] [ HA ] ) pH normal-p subscript K normal-a subscript 10 delimited-[] superscript normal-A delimited-[] HA \mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{[% \mathrm{HA}]}\right)

NTCIR12-MathWiki-3rate: 2

pH = p K a + log 10 ( [ A - ] [ HA ] ) pH normal-p subscript K normal-a subscript 10 delimited-[] superscript normal-A delimited-[] HA \mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{[% \mathrm{HA}]}\right)

NTCIR12-MathWiki-3rate: 2

log b ( x y ) = log b ( x ) - log b ( y ) subscript b x y subscript b x subscript b y \log_{b}\!\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)\,

NTCIR12-MathWiki-3rate: 1

f ( n ) = 2 ( n - 2 log 2 ( n ) ) + 1 f n 2 n superscript 2 subscript 2 n 1 f(n)=2(n-2^{\lfloor\log_{2}(n)\rfloor})+1

NTCIR12-MathWiki-3rate: 0

δ = 𝐡 ^ ( n ) - 𝐡 ( n ) δ normal-^ 𝐡 n 𝐡 n \mathbf{\delta}=\hat{\mathbf{h}}(n)-\mathbf{h}(n)

NTCIR12-MathWiki-3rate: 0

y = p x - f ( p ) y p x superscript f normal-⋆ p y=px-f^{\star}(p)

NTCIR12-MathWiki-3rate: 0

r = 0.5 + μ 2 , l = 4 r 0.5 subscript μ 2 l 4 r=\lfloor 0.5+\mu_{2,l}\rfloor=4

NTCIR12-MathWiki-3rate: 1

= log 2 ( depth ( v ) ) normal-ℓ subscript 2 depth v \ell=\lfloor\log_{2}(\operatorname{depth}(v))\rfloor

NTCIR12-MathWiki-3rate: 0

R T = f ( x ) - p ( x ) subscript R T f x p x R_{T}=f(x)-p(x)\,\!

NTCIR12-MathWiki-3rate: 1

2 - log 2 ( 2 ) = 3 2 2 subscript 2 2 3 2 2-\log_{2}(\sqrt{2})=\frac{3}{2}

NTCIR12-MathWiki-3rate: 1

- log ( 2 ) log ( 1 - γ 2 ) 2 1 γ 2 \frac{-\log(2)}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}

NTCIR12-MathWiki-3rate: 2

log b ( x y ) = log b ( x ) - log b ( y ) subscript b x y subscript b x subscript b y \log_{b}\!\left(\begin{matrix}\frac{x}{y}\end{matrix}\right)=\log_{b}(x)-\log_% {b}(y)

NTCIR12-MathWiki-3rate: 2

log 2 ( 1 2 ) = - 1 , subscript 2 1 2 1 \log_{2}\!\left(\frac{1}{2}\right)=-1,\,

NTCIR12-MathWiki-3rate: 1

log b ( x y ) = log b ( x ) - log b ( y ) subscript b x y subscript b x subscript b y \log_{b}\!\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)

NTCIR12-MathWiki-3rate: 2

m star = m - 2.5 log 10 [ L star L ( d d star ) 2 ] subscript m star subscript m direct-product 2.5 subscript 10 subscript L star subscript L direct-product superscript subscript d direct-product subscript d star 2 m_{\rm star}=m_{\odot}-2.5\log_{10}\left[\frac{L_{\rm star}}{L_{\odot}}\left(% \frac{d_{\odot}}{d_{\rm star}}\right)^{2}\right]

NTCIR12-MathWiki-3rate: 2

A = - log ( V d c V e x ) A subscript V d c subscript V e x A=-\log(\frac{V_{dc}}{V_{ex}})

NTCIR12-MathWiki-3rate: 2

M = log 2 ( R / G ) = log 2 ( R ) - log 2 ( G ) M subscript 2 R G subscript 2 R subscript 2 G M=\log_{2}(R/G)=\log_{2}(R)-\log_{2}(G)

NTCIR12-MathWiki-3rate: 1

[ Fe / H ] = log 10 ( N Fe N H ) star - log 10 ( N Fe N H ) sun fragments fragments normal-[ Fe H normal-] subscript 10 subscript fragments normal-( subscript N Fe subscript N normal-H normal-) star subscript 10 subscript fragments normal-( subscript N Fe subscript N normal-H normal-) sun [\mathrm{Fe}/\mathrm{H}]=\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}% }\right)_{\mathrm{star}}}-\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}% }}\right)_{\mathrm{sun}}}

NTCIR12-MathWiki-3rate: 3

d = 69 + 12 log 2 ( f 440 Hz ) . d 69 12 subscript 2 f 440 Hz d=69+12\log_{2}\left(\frac{f}{440\ \mathrm{Hz}}\right).\,

NTCIR12-MathWiki-3rate: 1

1 + log 2 N 1 subscript 2 N 1+\lfloor\log_{2}N\rfloor

NTCIR12-MathWiki-3rate: 2

L = log 2 n + 1 2 L subscript 2 n 1 2 L=\left\lceil\frac{\lfloor\log_{2}n\rfloor+1}{2}\right\rceil

NTCIR12-MathWiki-3rate: 0

0 = - d P d x + μ ( d u 2 d y 2 ) 0 d P d x μ d superscript u 2 d superscript y 2 0=-\frac{\mbox{d}~{}P}{\mbox{d}~{}x}+\mu\left(\frac{\mbox{d}~{}^{2}u}{\mbox{d}% ~{}y^{2}}\right)

NTCIR12-MathWiki-3rate: 0

α = arctan ( R L ) - arccos ( L 2 + R 2 2 ρ ) α R L superscript L 2 superscript R 2 2 ρ \alpha=\arctan\left({R\over L}\right)-\arccos\left({\sqrt{L^{2}+R^{2}}\over 2% \rho}\right)

NTCIR12-MathWiki-3rate: 0

( n - 1 ) m + 1 = ( 7 - 1 ) 4 + 1 = 6 4 + 1 = 1 + 1 = 2 n 1 m 1 7 1 4 1 6 4 1 1 1 2 \left\lfloor\frac{(n-1)}{m}\right\rfloor+1=\left\lfloor\frac{(7-1)}{4}\right% \rfloor+1=\left\lfloor\frac{6}{4}\right\rfloor+1=1+1=2

NTCIR12-MathWiki-3rate: 1

f ( n ) = n + log 2 ( n + log 2 n ) . f n n subscript 2 n subscript 2 n f(n)=n+\left\lfloor\log_{2}\left(n+\log_{2}n\right)\right\rfloor.

NTCIR12-MathWiki-3rate: 0

W C = U ( B ) - U ( A ) , subscript W C U B U A W_{C}=U(B)-U(A),

NTCIR12-MathWiki-3rate: 1

- log 10 ( 1 + 2 d 365 ) subscript 10 1 2 d 365 -\log_{10}\left(\frac{1+2d}{365}\right)

NTCIR12-MathWiki-3rate: 1

length ( c k ) - log 2 ( p k ) length subscript c k subscript 2 subscript p k \mathrm{length}(c_{k})\approx-\log_{2}\left(p_{k}\right)

NTCIR12-MathWiki-3rate: 1

Δ n ( s ) = { x n s } - log 2 ( 1 + s ) , fragments subscript normal-Δ n fragments normal-( s normal-) P fragments normal-{ subscript x n s normal-} subscript 2 fragments normal-( 1 s normal-) normal-, \Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s),

NTCIR12-MathWiki-3rate: 0

q = y - 0.5 = - - y + 0.5 q y 0.5 y 0.5 q=\left\lceil y-0.5\right\rceil=-\left\lfloor-y+0.5\right\rfloor\,

NTCIR12-MathWiki-3rate: 1

log 2 ( n ) subscript 2 n \lfloor\log_{2}(n)\rfloor

NTCIR12-MathWiki-3rate: 0

H H P F ( f ) = 1 - rect ( f 2 B H ) . subscript H H P F f 1 rect f 2 subscript B H H_{HPF}(f)=1-\mathrm{rect}\left(\frac{f}{2B_{H}}\right).

NTCIR12-MathWiki-3rate: 2

log ( x / y ) = log ( x ) - log ( y ) x y x y \log(x/y)=\log(x)-\log(y)

NTCIR12-MathWiki-3rate: 2

L N = 40 + 10 log 2 ( N ) subscript L N 40 10 subscript 2 N L_{N}=40+10\log_{2}(N)

NTCIR12-MathWiki-3rate: 2

L W = L p + 10 log 10 ( 4 π r 2 A 0 ) dB , subscript L W subscript L p 10 subscript 10 4 π superscript r 2 subscript A 0 dB L_{W}=L_{p}+10\log_{10}\!\left(\frac{4\pi r^{2}}{A_{0}}\right)\!~{}\mathrm{dB},

NTCIR12-MathWiki-3rate: 2

L p 2 = L p 1 + 20 log 10 ( r 1 r 2 ) dB . subscript L subscript p 2 subscript L subscript p 1 20 subscript 10 subscript r 1 subscript r 2 dB L_{p_{2}}=L_{p_{1}}+20\log_{10}\!\left(\frac{r_{1}}{r_{2}}\right)\!~{}\mathrm{% dB}.

NTCIR12-MathWiki-3rate: 0

e = H ( x ) - H ( x ^ ) e H x H normal-^ x e=H(x)-H(\hat{x})

NTCIR12-MathWiki-3rate: 0

e ˙ 2 = h 3 ( x ^ ) - m 2 ( x ^ ) sgn ( e 2 ) subscript normal-˙ e 2 subscript h 3 normal-^ x subscript m 2 normal-^ x sgn subscript e 2 \dot{e}_{2}=h_{3}(\hat{x})-m_{2}(\hat{x})\operatorname{sgn}(e_{2})

NTCIR12-MathWiki-3rate: 1

= lim N N b log 2 ( A N W + 1 ) absent subscript normal-→ N N b subscript 2 A N W 1 =\lim_{N\to\infty}Nb\log_{2}\left(\frac{A}{NW}+1\right)

NTCIR12-MathWiki-3rate: 1

γ 1 = ln 2 2 k = 2 ( - 1 ) k k log 2 k ( 2 log 2 k - log 2 2 k ) subscript γ 1 2 2 superscript subscript k 2 normal-⋅ superscript 1 k k subscript 2 k 2 subscript 2 k subscript 2 2 k \gamma_{1}\,=\,\frac{\ln 2}{2}\sum_{k=2}^{\infty}\frac{(-1)^{k}}{k}\,\lfloor% \log_{2}{k}\rfloor\cdot\big(2\log_{2}{k}-\lfloor\log_{2}{2k}\rfloor\big)

NTCIR12-MathWiki-3rate: 1

r = m s - deg ( f ) subscript r m s degree f r_{\infty}=ms-\deg(f)

NTCIR12-MathWiki-3rate: 2

1 + log 2 n 1 subscript 2 n 1+\lfloor\log_{2}n\rfloor

NTCIR12-MathWiki-3rate: 0

c 2 I = c 1 E - S e - 1 ( E k e - r e E ) + P subscript c 2 I subscript c 1 E superscript subscript S e 1 E subscript k e subscript r e E P c_{2}I=c_{1}E-S_{e}^{-1}\left(\frac{E}{k_{e}-r_{e}E}\right)+P

NTCIR12-MathWiki-3rate: 4

N = 0.5 - log 2 ( Frequency of this item Frequency of most common item ) N 0.5 subscript 2 Frequency of this item Frequency of most common item N=\left\lfloor 0.5-\log_{2}\left(\frac{\,\text{Frequency of this item}}{\,% \text{Frequency of most common item}}\right)\right\rfloor

NTCIR12-MathWiki-3rate: 0

N = u 2 N superscript u 2 N=\lfloor u^{2}\rfloor

NTCIR12-MathWiki-4rate: 1

t ( ρ 0 + ρ 0 s ) + x ( ρ 0 u + ρ 0 s u ) = 0 t subscript ρ 0 subscript ρ 0 s x subscript ρ 0 u subscript ρ 0 s u 0 \frac{\partial}{\partial t}(\rho_{0}+\rho_{0}s)+\frac{\partial}{\partial x}(% \rho_{0}u+\rho_{0}su)=0

NTCIR12-MathWiki-4rate: 2

× 𝐁 = μ 0 𝐉 normal-∇ 𝐁 subscript μ 0 𝐉 \mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}

NTCIR12-MathWiki-4rate: 2

× 𝐁 = μ 0 𝐉 normal-∇ 𝐁 subscript μ 0 𝐉 \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}

NTCIR12-MathWiki-4rate: 0

i t n 𝐤 c = ( Ω 𝐤 p 𝐤 - Ω 𝐤 p 𝐤 ) + i t n 𝐤 c | corr , normal-i Planck-constant-over-2-pi t subscript superscript n c 𝐤 superscript subscript normal-Ω 𝐤 normal-⋆ subscript p 𝐤 subscript normal-Ω 𝐤 superscript subscript p 𝐤 normal-⋆ evaluated-at normal-i Planck-constant-over-2-pi t subscript superscript n c 𝐤 corr \mathrm{i}\hbar\frac{\partial}{\partial t}n^{c}_{\mathbf{k}}=(\Omega_{\mathbf{% k}}^{\star}\,p_{\mathbf{k}}-\Omega_{\mathbf{k}}\,p_{\mathbf{k}}^{\star})+% \mathrm{i}\hbar\frac{\partial}{\partial t}n^{c}_{\mathbf{k}}|_{\,\text{corr}}\,,

NTCIR12-MathWiki-4rate: 2

s y m b o l × ( s y m b o l × B ) = μ 0 ϵ 0 t s y m b o l × E . s y m b o l normal-∇ s y m b o l normal-∇ B subscript μ 0 subscript ϵ 0 t s y m b o l normal-∇ E symbol{\nabla\times}\left(symbol{\nabla\times B}\right)=\mu_{0}\epsilon_{0}% \frac{\partial}{\partial t}symbol{\nabla\times E}\ .

NTCIR12-MathWiki-4rate: 3

× 𝐁 = μ 0 𝐉 + μ 0 ε 0 𝐄 t normal-∇ 𝐁 subscript μ 0 𝐉 subscript μ 0 subscript ε 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}

NTCIR12-MathWiki-4rate: 1

2 𝐄 = μ 0 ϵ 0 2 𝐄 t 2 superscript normal-∇ 2 𝐄 subscript μ 0 subscript ϵ 0 superscript 2 𝐄 superscript t 2 \nabla^{2}\mathbf{E}=\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t% ^{2}}

NTCIR12-MathWiki-4rate: 1

× 𝐄 = 𝐤 ^ × 𝐄 0 f ( 𝐤 ^ 𝐱 - c 0 t ) = - 𝐁 t normal-∇ 𝐄 normal-^ 𝐤 subscript 𝐄 0 superscript f normal-′ normal-⋅ normal-^ 𝐤 𝐱 subscript c 0 t 𝐁 t \nabla\times\mathbf{E}=\hat{\mathbf{k}}\times\mathbf{E}_{0}f^{\prime}\left(% \hat{\mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)=-\frac{\partial\mathbf{B}}{% \partial t}

NTCIR12-MathWiki-4rate: 2

× 𝐁 = μ 0 ϵ 0 𝐄 t ( 4 ) normal-∇ 𝐁 subscript μ 0 subscript ϵ 0 𝐄 t 4 \nabla\times\mathbf{B}=\mu_{0}\epsilon_{0}\frac{\partial\mathbf{E}}{\partial t% }\qquad\quad\ (4)

NTCIR12-MathWiki-4rate: 3

𝐄 = ρ ϵ 0 , × 𝐁 - 1 c 2 𝐄 t = μ 0 𝐉 formulae-sequence normal-⋅ normal-∇ 𝐄 ρ subscript ϵ 0 normal-∇ 𝐁 1 superscript c 2 𝐄 t subscript μ 0 𝐉 \nabla\cdot\mathbf{E}=\frac{\rho}{\epsilon_{0}},\quad\nabla\times\mathbf{B}-% \frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}=\mu_{0}\mathbf{J}

NTCIR12-MathWiki-4rate: 2

× 𝐁 = μ 0 𝐣 normal-∇ 𝐁 subscript μ 0 𝐣 \nabla\times\mathbf{B}=\mu_{0}\mathbf{j}

NTCIR12-MathWiki-4rate: 1

𝐁 = μ 0 ρ m normal-⋅ normal-∇ 𝐁 subscript μ 0 subscript ρ m \nabla\cdot\mathbf{B}=\mu_{0}\rho_{m}

NTCIR12-MathWiki-4rate: 3

× 𝐁 = μ 0 𝐉 + 1 c 2 𝐄 t normal-∇ 𝐁 subscript μ 0 𝐉 1 superscript c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}

NTCIR12-MathWiki-4rate: 3

× 𝐁 = 1 ϵ 0 c 2 𝐉 + 1 c 2 𝐄 t normal-∇ 𝐁 1 subscript ϵ 0 superscript c 2 𝐉 1 superscript c 2 𝐄 t \nabla\times\mathbf{B}=\frac{1}{\epsilon_{0}c^{2}}\mathbf{J}+\frac{1}{c^{2}}% \frac{\partial\mathbf{E}}{\partial t}

NTCIR12-MathWiki-4rate: 0

+ E sig E LO cos ( ( ω sig - ω LO ) t + φ ) b e a t c o m p o n e n t . subscript normal-⏟ subscript E sig subscript E LO subscript ω sig subscript ω LO t φ b e a t c o m p o n e n t +\underbrace{E_{\mathrm{sig}}E_{\mathrm{LO}}\cos((\omega_{\mathrm{sig}}-\omega% _{\mathrm{LO}})t+\varphi)}_{beat\;component}.

NTCIR12-MathWiki-4rate: 2

× B = μ 0 J , normal-→ normal-∇ normal-→ B subscript μ 0 normal-→ J \vec{\nabla}\times\vec{B}=\mu_{0}\vec{J},

NTCIR12-MathWiki-4rate: 1

2 φ + t ( 𝐀 ) = - ρ ε 0 , superscript normal-∇ 2 φ t normal-⋅ normal-∇ 𝐀 ρ subscript ε 0 \nabla^{2}\varphi+{{\partial}\over\partial t}\left(\nabla\cdot\mathbf{A}\right% )=-{\rho\over\varepsilon_{0}}\,,

NTCIR12-MathWiki-4rate: 4

× 𝐁 = μ 0 ( 𝐉 + ε 0 𝐄 t ) normal-∇ 𝐁 subscript μ 0 𝐉 subscript ε 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{J}+\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}\right)

NTCIR12-MathWiki-4rate: 3

× 𝐁 = μ 0 𝐉 + 1 c 2 𝐄 t normal-∇ 𝐁 subscript μ 0 𝐉 1 superscript c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}

NTCIR12-MathWiki-4rate: 2

E = ρ ϵ 0 × B - ϵ 0 μ 0 E t = μ 0 J × E + B t = 0 B = 0 formulae-sequence normal-⋅ normal-∇ normal-→ E ρ subscript ϵ 0 formulae-sequence normal-∇ normal-→ B subscript ϵ 0 subscript μ 0 normal-→ E t subscript μ 0 normal-→ J formulae-sequence normal-∇ normal-→ E normal-→ B t 0 normal-⋅ normal-∇ normal-→ B 0 \nabla\cdot\vec{E}={\rho\over\epsilon_{0}}\qquad\nabla\times\vec{B}-\epsilon_{% 0}\mu_{0}{\partial\vec{E}\over\partial t}=\mu_{0}\vec{J}\qquad\nabla\times\vec% {E}+{\partial\vec{B}\over\partial t}=0\qquad\nabla\cdot\vec{B}=0

NTCIR12-MathWiki-4rate: 3

× 𝐁 = μ 0 𝐉 + 1 c 2 𝐄 t normal-∇ 𝐁 subscript μ 0 𝐉 1 superscript c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{% E}}{\partial t}

NTCIR12-MathWiki-4rate: 3

× 𝐁 = μ 0 𝐉 + μ 0 ε 0 𝐄 t , normal-∇ 𝐁 subscript μ 0 𝐉 subscript μ 0 subscript ε 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t},

NTCIR12-MathWiki-4rate: 2

× 𝐁 = μ 0 𝐉 normal-∇ 𝐁 subscript μ 0 𝐉 \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}

NTCIR12-MathWiki-4rate: 4

× 𝐁 = μ 𝐉 + μ ϵ 𝐄 t . normal-∇ 𝐁 μ 𝐉 μ ϵ 𝐄 t \nabla\times\mathbf{B}=\mu\mathbf{J}+\mu\epsilon\frac{\partial\mathbf{E}}{% \partial t}.

NTCIR12-MathWiki-4rate: 1

𝐟 + ϵ 0 μ 0 𝐒 t = σ 𝐟 subscript ϵ 0 subscript μ 0 𝐒 t normal-⋅ normal-∇ σ \mathbf{f}+\epsilon_{0}\mu_{0}\frac{\partial\mathbf{S}}{\partial t}\,=\nabla% \cdot\mathbf{\sigma}

NTCIR12-MathWiki-4rate: 2

μ 0 𝐉 = × 𝐁 subscript μ 0 𝐉 normal-∇ 𝐁 \mu_{0}\mathbf{J}=\nabla\times\mathbf{B}

NTCIR12-MathWiki-4rate: 2

μ 0 𝐉 = × 𝐁 . subscript μ 0 𝐉 normal-∇ 𝐁 \mu_{0}\mathbf{J}=\nabla\times\mathbf{B}.

NTCIR12-MathWiki-4rate: 3

𝐀 = - μ 0 ε 0 φ t normal-⋅ normal-∇ superscript 𝐀 normal-′ subscript μ 0 subscript ε 0 superscript φ normal-′ t \mathbf{\nabla}\cdot\mathbf{A}^{\prime}=-\mu_{0}\varepsilon_{0}\frac{\partial% \varphi^{\prime}}{\partial t}

NTCIR12-MathWiki-4rate: 4

× 𝐁 = μ 0 𝐉 + μ 0 ε 0 𝐄 t normal-∇ 𝐁 subscript μ 0 𝐉 subscript μ 0 subscript ε 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}

NTCIR12-MathWiki-4rate: 3

× 𝐁 = μ 0 ( 𝐉 + ε 0 𝐄 t ) normal-∇ 𝐁 subscript μ 0 𝐉 subscript ε 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{J}+\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}\right)

NTCIR12-MathWiki-4rate: 4

× 𝐁 - 1 c 2 𝐄 t = μ 0 𝐉 normal-∇ 𝐁 1 superscript c 2 𝐄 t subscript μ 0 𝐉 \nabla\times\mathbf{B}-\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}=% \mu_{0}\mathbf{J}

NTCIR12-MathWiki-4rate: 3

𝐟 + ϵ 0 μ 0 𝐒 t = σ 𝐟 subscript ϵ 0 subscript μ 0 𝐒 t normal-⋅ normal-∇ σ \mathbf{f}+\epsilon_{0}\mu_{0}\frac{\partial\mathbf{S}}{\partial t}\,=\nabla% \cdot\mathbf{\sigma}

NTCIR12-MathWiki-4rate: 4

× 𝐁 = μ 0 𝐉 + μ 0 ϵ 0 𝐄 t normal-∇ 𝐁 subscript μ 0 𝐉 subscript μ 0 subscript ϵ 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\epsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}

NTCIR12-MathWiki-4rate: 4

× 𝐁 = μ 0 𝐉 + μ 0 ϵ 0 t 𝐄 Maxwell s term normal-∇ 𝐁 subscript μ 0 𝐉 subscript normal-⏟ subscript μ 0 subscript ϵ 0 t 𝐄 superscript Maxwell normal-′ normal-s term \mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}+\underbrace{\mu_{0}\epsilon_% {0}\frac{\partial}{\partial t}\mathbf{E}}_{\mathrm{Maxwell^{\prime}s\ term}}

NTCIR12-MathWiki-4rate: 2

× B = μ 0 J normal-∇ normal-→ B subscript μ 0 normal-→ J \nabla\times\vec{B}=\mu_{0}\vec{J}

NTCIR12-MathWiki-4rate: 2

× B = μ 0 J normal-∇ normal-→ B subscript μ 0 normal-→ J \nabla\times\vec{B}=\mu_{0}\vec{J}

NTCIR12-MathWiki-4rate: 2

× B = μ 0 J normal-∇ normal-→ B subscript μ 0 normal-→ J \nabla\times\vec{B}=\mu_{0}\vec{J}

NTCIR12-MathWiki-4rate: 3

× 𝐁 = 1 c 2 ( 1 ϵ 0 𝐉 + 𝐄 t ) normal-∇ 𝐁 1 superscript c 2 1 subscript ϵ 0 𝐉 𝐄 t \nabla\times\mathbf{B}=\frac{1}{c^{2}}\left(\frac{1}{\epsilon_{0}}\mathbf{J}+% \frac{\partial\mathbf{E}}{\partial t}\right)

NTCIR12-MathWiki-4rate: 1

𝐒 + ϵ 0 𝐄 𝐄 t + 𝐁 μ 0 𝐁 t + 𝐉 𝐄 = 0 , normal-⋅ normal-∇ 𝐒 normal-⋅ subscript ϵ 0 𝐄 𝐄 t normal-⋅ 𝐁 subscript μ 0 𝐁 t normal-⋅ 𝐉 𝐄 0 \nabla\cdot\mathbf{S}+\epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{% \partial t}+\frac{\mathbf{B}}{\mu_{0}}\cdot\frac{\partial\mathbf{B}}{\partial t% }+\mathbf{J}\cdot\mathbf{E}=0,

NTCIR12-MathWiki-4rate: 0

ϵ 0 𝐄 𝐄 t normal-⋅ subscript ϵ 0 𝐄 𝐄 t \epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{\partial t}

NTCIR12-MathWiki-4rate: 0

2 E - μ 0 ε 0 2 E t 2 = 0. superscript normal-∇ 2 E subscript μ 0 subscript ε 0 superscript 2 E superscript t 2 0. \nabla^{2}E-\mu_{0}\,\varepsilon_{0}\,\frac{\partial^{2}E}{\partial t^{2}}=0.

NTCIR12-MathWiki-4rate: 0

I = τ - 0 H ( μ + τ x ) e x + 1 d x I 1 + τ 0 H ( μ + τ x ) e x + 1 d x I 2 . I subscript normal-⏟ τ superscript subscript 0 H μ τ x superscript e x 1 normal-d x subscript I 1 subscript normal-⏟ τ superscript subscript 0 H μ τ x superscript e x 1 normal-d x subscript I 2 I=\underbrace{\tau\int_{-\infty}^{0}\frac{H(\mu+\tau x)}{e^{x}+1}\,\mathrm{d}x% }_{I_{1}}+\underbrace{\tau\int_{0}^{\infty}\frac{H(\mu+\tau x)}{e^{x}+1}\,% \mathrm{d}x}_{I_{2}}\,.

NTCIR12-MathWiki-4rate: 3

× 𝐁 = μ 0 ε 0 𝐄 t = 1 c 2 𝐄 t normal-∇ 𝐁 subscript μ 0 subscript ε 0 𝐄 t 1 superscript c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{% \partial t}=\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}

NTCIR12-MathWiki-4rate: 0

h t Change in mass over time + h u ¯ x + h v ¯ y Total spatial variation of x,y mass fluxes = 0 subscript normal-⏟ h t Change in mass over time subscript normal-⏟ normal-¯ h u x normal-¯ h v y Total spatial variation of x,y mass fluxes 0 {\underbrace{\partial h\over\partial t}}_{\begin{smallmatrix}\,\text{Change}\\ \,\text{in mass}\\ \,\text{over time}\end{smallmatrix}}+\underbrace{{\partial\overline{hu}\over% \partial x}+{\partial\overline{hv}\over\partial y}}_{\begin{smallmatrix}\,% \text{Total spatial}\\ \,\text{variation of}\\ \,\text{x,y mass fluxes}\end{smallmatrix}}=0

NTCIR12-MathWiki-4rate: 2

× 𝐁 = μ 0 𝐣 normal-∇ 𝐁 subscript μ 0 𝐣 \nabla\times\mathbf{B}=\mu_{0}\mathbf{j}

NTCIR12-MathWiki-4rate: 2

2 𝐄 - μ 0 ϵ 0 2 𝐄 t 2 = 0. superscript normal-∇ 2 𝐄 subscript μ 0 subscript ϵ 0 superscript 2 𝐄 superscript t 2 0. \nabla^{2}\mathbf{E}-\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t% ^{2}}=0.

NTCIR12-MathWiki-4rate: 1

× 𝐁 ^ = μ 0 𝐉 ^ + i μ 0 ε 0 ω 𝐄 ^ - B r r + d B ( 1 ) d r + B ( 1 ) r = μ 0 J + i ω μ 0 ε 0 E formulae-sequence normal-∇ normal-^ 𝐁 subscript μ 0 normal-^ 𝐉 normal-i subscript μ 0 subscript ε 0 ω normal-^ 𝐄 normal-⇒ superscript B r r normal-d superscript B 1 normal-d r superscript B 1 r subscript μ 0 J normal-i ω subscript μ 0 subscript ε 0 E \nabla\times\hat{\mathbf{B}}=\mu_{0}\hat{\mathbf{J}}+\mathrm{i}\mu_{0}% \varepsilon_{0}\omega\hat{\mathbf{E}}\quad\Rightarrow\quad-\frac{B^{r}}{r}+% \frac{\mathrm{d}B^{(1)}}{\mathrm{d}r}+\frac{B^{(1)}}{r}=\mu_{0}J+\mathrm{i}% \omega\mu_{0}\varepsilon_{0}E

NTCIR12-MathWiki-4rate: 2

× 𝐁 𝟏 = μ 0 𝐉 𝟏 - i ω ϵ 0 μ 0 𝐄 𝟏 normal-∇ subscript 𝐁 1 subscript μ 0 subscript 𝐉 1 i ω subscript ϵ 0 subscript μ 0 subscript 𝐄 1 \nabla\times\mathbf{B_{1}}=\mu_{0}\mathbf{J_{1}}-i\omega\epsilon_{0}\mu_{0}% \mathbf{E_{1}}

NTCIR12-MathWiki-5rate: 1

1 + 1 4 + 1 1 + 1 18 + 1 1 1 4 continued-fraction 1 1 continued-fraction 1 18 continued-fraction 1 fragments normal-⋱ italic- 1+\frac{1}{4+\cfrac{1}{1+\cfrac{1}{18+\cfrac{1}{\ddots\qquad{}}}}}

NTCIR12-MathWiki-5rate: 1

π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + π continued-fraction 4 1 continued-fraction superscript 1 2 3 continued-fraction superscript 2 2 5 continued-fraction superscript 3 2 7 normal-⋱ \pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}\!

NTCIR12-MathWiki-5rate: 4

1 + 1 2 + 1 5 + 1 5 + 1 4 + 1 continued-fraction 1 2 continued-fraction 1 5 continued-fraction 1 5 continued-fraction 1 4 normal-⋱ 1+\cfrac{1}{2+\cfrac{1}{5+\cfrac{1}{5+\cfrac{1}{4+\ddots}}}}

NTCIR12-MathWiki-5rate: 1

1 3 + 1 1 + 1 1 + 1 3 + 1 9 + continued-fraction 1 3 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 3 continued-fraction 1 9 normal-⋱ \cfrac{1}{3+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{9+\ddots}}}}}

NTCIR12-MathWiki-5rate: 2

π = 3 + 1 7 + 1 15 + 1 1 + 1 292 + 1 1 + 1 1 + 1 1 + 1 2 + 1 1 + 1 3 + 1 1 + π 3 continued-fraction 1 7 continued-fraction 1 15 continued-fraction 1 1 continued-fraction 1 292 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 1 continued-fraction 1 3 continued-fraction 1 1 normal-⋱ \pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}% {1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\ddots}}}}}}}}}}}

NTCIR12-MathWiki-5rate: 1

π = 2 + 4 3 + 1 3 4 + 3 5 4 + 5 7 4 + π 2 continued-fraction 4 3 continued-fraction normal-⋅ 1 3 4 continued-fraction normal-⋅ 3 5 4 continued-fraction normal-⋅ 5 7 4 normal-⋱ \displaystyle\pi=2+\cfrac{4}{3+\cfrac{1\cdot 3}{4+\cfrac{3\cdot 5}{4+\cfrac{5% \cdot 7}{4+\ddots}}}}

NTCIR12-MathWiki-5rate: 1

π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + . π continued-fraction 4 1 continued-fraction superscript 1 2 2 continued-fraction superscript 3 2 2 continued-fraction superscript 5 2 2 continued-fraction superscript 7 2 2 normal-⋱ \pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\ddots}}}}}.\,

NTCIR12-MathWiki-5rate: 1

e 2 = 1 + 4 0 + 2 2 6 + 2 2 10 + 2 2 14 + = 7 + 2 5 + 1 7 + 1 9 + 1 11 + superscript e 2 1 continued-fraction 4 0 continued-fraction superscript 2 2 6 continued-fraction superscript 2 2 10 continued-fraction superscript 2 2 14 normal-⋱ 7 continued-fraction 2 5 continued-fraction 1 7 continued-fraction 1 9 continued-fraction 1 11 normal-⋱ e^{2}=1+\cfrac{4}{0+\cfrac{2^{2}}{6+\cfrac{2^{2}}{10+\cfrac{2^{2}}{14+\ddots\,% }}}}=7+\cfrac{2}{5+\cfrac{1}{7+\cfrac{1}{9+\cfrac{1}{11+\ddots\,}}}}

NTCIR12-MathWiki-5rate: 2

e = 2 + 1 1 + 1 𝟐 + 1 1 + 1 1 + 1 𝟒 + 1 1 + 1 1 + = 1 + 1 𝟎 + 1 1 + 1 1 + 1 𝟐 + 1 1 + 1 1 + 1 𝟒 + 1 1 + 1 1 + . e 2 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 4 continued-fraction 1 1 continued-fraction 1 1 normal-⋱ 1 continued-fraction 1 0 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 4 continued-fraction 1 1 continued-fraction 1 1 normal-⋱ e=2+\cfrac{1}{1+\cfrac{1}{\mathbf{2}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\mathbf% {4}+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}}}}}=1+\cfrac{1}{\mathbf{0}+\cfrac{1}{1+% \cfrac{1}{1+\cfrac{1}{\mathbf{2}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\mathbf{4}+% \cfrac{1}{1+\cfrac{1}{1+\ddots}}}}}}}}}.

NTCIR12-MathWiki-5rate: 1

e = 1 + 2 1 + 1 6 + 1 10 + 1 14 + 1 18 + 1 22 + 1 26 + . e 1 continued-fraction 2 1 continued-fraction 1 6 continued-fraction 1 10 continued-fraction 1 14 continued-fraction 1 18 continued-fraction 1 22 continued-fraction 1 26 normal-⋱ e=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\cfrac{1}{2% 2+\cfrac{1}{26+\ddots\,}}}}}}}.

NTCIR12-MathWiki-5rate: 1

φ = 1 + 1 1 + 1 1 + 1 1 + φ 1 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 1 normal-⋱ \varphi=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\;\;\ddots\,}}}

NTCIR12-MathWiki-5rate: 2

e = 2 + 1 1 + 1 2 + 1 1 + 1 1 + 1 4 + e 2 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 4 normal-⋱ e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+{}\ddots}}}}}

NTCIR12-MathWiki-5rate: 3

x = 1 + 1 1 + 1 1 + 1 1 + 1 1 + x 1 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 1 normal-⋱ x=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots\,}}}}

NTCIR12-MathWiki-5rate: 1

log 2 = log ( 1 + 1 ) = 1 1 + 1 2 + 1 3 + 2 2 + 2 5 + 3 2 + = 2 3 - 1 2 9 - 2 2 15 - 3 2 21 - 2 1 1 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 3 continued-fraction 2 2 continued-fraction 2 5 continued-fraction 3 2 normal-⋱ continued-fraction 2 3 continued-fraction superscript 1 2 9 continued-fraction superscript 2 2 15 continued-fraction superscript 3 2 21 normal-⋱ \log 2=\log(1+1)=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+% \cfrac{3}{2+\ddots}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2}}{15-\cfrac{3% ^{2}}{21-\ddots}}}}

NTCIR12-MathWiki-5rate: 2

1 + 1 1 + 1 1 + 1 1 + 1 1 + 1 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 1 normal-⋱ 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}}

NTCIR12-MathWiki-5rate: 4

1 + 1 1 + 1 1 + 1 5 + 1 1 + 1 4 + 1 continued-fraction 1 1 continued-fraction 1 1 continued-fraction 1 5 continued-fraction 1 1 continued-fraction 1 4 normal-⋱ 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{5+\cfrac{1}{1+\cfrac{1}{4+\ddots}}}}}

NTCIR12-MathWiki-5rate: 1

π = 3 + 1 2 6 + 3 2 6 + 5 2 6 + 7 2 6 + π 3 continued-fraction superscript 1 2 6 continued-fraction superscript 3 2 6 continued-fraction superscript 5 2 6 continued-fraction superscript 7 2 6 normal-⋱ \pi={3+\cfrac{1^{2}}{6+\cfrac{3^{2}}{6+\cfrac{5^{2}}{6+\cfrac{7^{2}}{6+\ddots% \,}}}}}

NTCIR12-MathWiki-5rate: 1

π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + 4 2 9 + π continued-fraction 4 1 continued-fraction superscript 1 2 3 continued-fraction superscript 2 2 5 continued-fraction superscript 3 2 7 continued-fraction superscript 4 2 9 normal-⋱ \pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\cfrac{4^{2}}{% 9+\ddots}}}}}

NTCIR12-MathWiki-5rate: 1

π = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + π continued-fraction 4 1 continued-fraction superscript 1 2 2 continued-fraction superscript 3 2 2 continued-fraction superscript 5 2 2 continued-fraction superscript 7 2 2 normal-⋱ \pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\ddots}}}}}\,

NTCIR12-MathWiki-5rate: 2

e = 2 + 1 1 + 1 2 + 2 3 + 3 4 + 4 5 + = 2 + 2 2 + 3 3 + 4 4 + 5 5 + 6 6 + e 2 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 2 3 continued-fraction 3 4 continued-fraction 4 5 normal-⋱ 2 continued-fraction 2 2 continued-fraction 3 3 continued-fraction 4 4 continued-fraction 5 5 continued-fraction 6 6 normal-⋱ e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}}=2+% \cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}

NTCIR12-MathWiki-5rate: 1

1 1 + 1 0 + 1 8 + 1 4 + 1 1 + 1 0 + 1 / 1 1 1 0 1 8 1 4 1 1 1 0 1 normal-⋯ \tfrac{1}{1+\tfrac{1}{0+\tfrac{1}{8+\tfrac{1}{4+\tfrac{1}{1+\tfrac{1}{0+1{/% \cdots}}}}}}}

NTCIR12-MathWiki-5rate: 2

ln 2 = 1 1 + 1 2 + 1 3 + 2 2 + 2 5 + 3 2 + 3 7 + 4 2 + = 2 3 - 1 2 9 - 2 2 15 - 3 2 21 - 2 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 3 continued-fraction 2 2 continued-fraction 2 5 continued-fraction 3 2 continued-fraction 3 7 continued-fraction 4 2 normal-⋱ continued-fraction 2 3 continued-fraction superscript 1 2 9 continued-fraction superscript 2 2 15 continued-fraction superscript 3 2 21 normal-⋱ \ln 2=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+\cfrac{3}{2+% \cfrac{3}{7+\cfrac{4}{2+\ddots}}}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2% }}{15-\cfrac{3^{2}}{21-\ddots}}}}

NTCIR12-MathWiki-5rate: 1

2 = 1 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + . 2 1 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 normal-⋱ \sqrt{2}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots% \,}}}}}.

NTCIR12-MathWiki-5rate: 1

577 408 = 1 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 . 577 408 1 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 \frac{577}{408}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+% \cfrac{1}{2+\cfrac{1}{2}}}}}}}.

NTCIR12-MathWiki-5rate: 1

π = 3 + 1 7 + 1 15 + 1 1 + 1 292 + 1 1 + 1 1 + 1 1 + π 3 1 7 1 15 1 1 1 292 1 1 1 1 1 1 normal-⋱ \pi=3+\textstyle\frac{1}{7+\textstyle\frac{1}{15+\textstyle\frac{1}{1+% \textstyle\frac{1}{292+\textstyle\frac{1}{1+\textstyle\frac{1}{1+\textstyle% \frac{1}{1+\ddots}}}}}}}

NTCIR12-MathWiki-5rate: 1

a 0 + 1 a 1 + 1 a 2 + 1 + 1 a n , subscript a 0 continued-fraction 1 subscript a 1 continued-fraction 1 subscript a 2 continued-fraction 1 normal-⋱ continued-fraction 1 subscript a n a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots+\cfrac{1}{a_{n}}}}},

NTCIR12-MathWiki-5rate: 1

2 + 1 2 + 1 2 + 1 2 + 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 normal-⋱ 2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}

NTCIR12-MathWiki-5rate: 1

tanh 1 2 = e - 1 e + 1 = 0 + 1 2 + 1 6 + 1 10 + 1 14 + 1 1 2 e 1 e 1 0 continued-fraction 1 2 continued-fraction 1 6 continued-fraction 1 10 continued-fraction 1 14 continued-fraction 1 normal-⋱ \tanh\frac{1}{2}=\frac{e-1}{e+1}=0+\cfrac{1}{2+\cfrac{1}{6+\cfrac{1}{10+\cfrac% {1}{14+\cfrac{1}{\ddots}}}}}

NTCIR12-MathWiki-5rate: 2

x = 1 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + = 2 . x 1 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 normal-⋱ 2 x=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}}=% \sqrt{2}.\,

NTCIR12-MathWiki-5rate: 1

11 = 3 + 1 3 + 1 6 + 1 3 + 1 6 + 1 3 + 11 3 continued-fraction 1 3 continued-fraction 1 6 continued-fraction 1 3 continued-fraction 1 6 continued-fraction 1 3 normal-⋱ \sqrt{11}=3+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\ddots% }}}}}

NTCIR12-MathWiki-5rate: 1

1 + 1 2 + 1 2 + 1 2 + 1 2 + 1 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 continued-fraction 1 2 normal-⋱ 1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}

NTCIR12-MathWiki-5rate: 2

1 + 1 1 + 1 2 + 1 1 + 1 2 + 1 1 + 1 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 1 continued-fraction 1 2 continued-fraction 1 1 normal-⋱ 1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\ddots}}}}}

NTCIR12-MathWiki-5rate: 1

2 + 1 4 + 1 4 + 1 4 + 1 4 + 2 continued-fraction 1 4 continued-fraction 1 4 continued-fraction 1 4 continued-fraction 1 4 normal-⋱ 2+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\ddots}}}}

NTCIR12-MathWiki-5rate: 3

1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 continued-fraction 1 2 continued-fraction 1 3 continued-fraction 1 4 continued-fraction 1 5 continued-fraction 1 6 normal-⋱ {1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\ddots}}}}}}

NTCIR12-MathWiki-5rate: 1

4 π = 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + 4 π 1 continued-fraction superscript 1 2 2 continued-fraction superscript 3 2 2 continued-fraction superscript 5 2 2 continued-fraction superscript 7 2 2 continued-fraction superscript 9 2 2 normal-⋱ \frac{4}{\pi}=1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{% 2+\cfrac{9^{2}}{2+\ddots}}}}}

NTCIR12-MathWiki-6rate: 2

U 92 238 + n 0 1 U 92 239 β - 23.5 min Np 93 239 β - 2.3 days Pu 94 239 2.4 10 4 years 𝛼 normal-→ superscript subscript normal-U 92 238 superscript subscript normal-n 0 1 superscript subscript normal-U 92 239 23.5 min superscript β normal-→ superscript subscript Np 93 239 2.3 days superscript β normal-→ superscript subscript Pu 94 239 normal-⋅ 2.4 superscript 10 4 years α normal-→ absent \mathrm{{}^{238}_{92}U+{}^{1}_{0}n\ \xrightarrow{\ }\ {}^{239}_{92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ {}^{239}_{93}Np\ \xrightarrow[2.3\ days]{% \beta^{-}}\ {}^{239}_{94}Pu\ \xrightarrow[2.4\cdot 10^{4}\ years]{\alpha}}

NTCIR12-MathWiki-6rate: 2

Ra 88 226 + 0 1 n 88 227 Ra β - 42.2 min 89 227 Ac subscript superscript normal-⟶ 227 88 subscript superscript 1 0 superscript subscript Ra 88 226 normal-n Ra subscript superscript 42.2 min superscript β normal-→ 227 89 Ac \mathrm{{}^{226}_{\ 88}Ra\ +\ ^{1}_{0}n\ \longrightarrow\ ^{227}_{\ 88}Ra\ % \xrightarrow[42.2\ min]{\beta^{-}}\ ^{227}_{\ 89}Ac}

NTCIR12-MathWiki-6rate: 4

U 92 238 Th 90 234 + α normal-→ superscript subscript normal-U 92 238 superscript subscript Th 90 234 α \mathrm{~{}^{238}_{92}U}\rightarrow\mathrm{~{}^{234}_{90}Th}+{\alpha}

NTCIR12-MathWiki-6rate: 2

Ca 20 40 + He 2 4 Ti 22 44 + γ normal-→ subscript superscript Ca 40 20 subscript superscript He 4 2 subscript superscript Ti 44 22 γ \mathrm{{}_{20}^{40}Ca}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{22}^{44}Ti% }+\gamma

NTCIR12-MathWiki-6rate: 2

U 92 238 ( n , γ ) 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu subscript superscript normal-n γ normal-→ 239 92 superscript subscript normal-U 92 238 normal-U subscript superscript 23.5 min superscript β normal-→ 239 93 Np subscript superscript 2.3565 normal-d superscript β normal-→ 239 94 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 0

B + X Y + D normal-→ B X Y D B+X\rightarrow Y+D

NTCIR12-MathWiki-6rate: 2

U 92 238 + 5 10 B 97 242 Bk + 6 0 1 n ; 90 232 Th + 7 15 N 97 242 Bk + 5 0 1 n fragments superscript subscript normal-U 92 238 subscript superscript 10 5 B subscript superscript normal-⟶ 242 97 Bk subscript superscript 6 1 0 n subscript superscript normal-; 232 90 Th subscript superscript 15 7 N subscript superscript normal-⟶ 242 97 Bk subscript superscript 5 1 0 n \mathrm{{}^{238}_{\ 92}U\ +\ ^{10}_{\ 5}B\ \longrightarrow\ ^{242}_{\ 97}Bk\ +% \ 6\ ^{1}_{0}n\quad;\quad^{232}_{\ 90}Th\ +\ ^{15}_{\ 7}N\ \longrightarrow\ ^{% 242}_{\ 97}Bk\ +\ 5\ ^{1}_{0}n}

NTCIR12-MathWiki-6rate: 2

U 92 238 ( n , γ ) 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu subscript superscript normal-n γ normal-→ 239 92 superscript subscript normal-U 92 238 normal-U subscript superscript 23.5 min superscript β normal-→ 239 93 Np subscript superscript 2.3565 normal-d superscript β normal-→ 239 94 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 1

Glucose + Alkaline copper tartarate Reduction Cuprous oxide Reduction normal-→ Glucose Alkaline copper tartarate Cuprous oxide \mathrm{Glucose}+\mathrm{Alkaline\ copper\ tartarate}\xrightarrow{\mathrm{% Reduction}}\mathrm{Cuprous\ oxide}

NTCIR12-MathWiki-6rate: 0

B + X Y + D normal-→ B X Y D B+X\rightarrow Y+D

NTCIR12-MathWiki-6rate: 1

U 92 238 ( n , γ ) 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu subscript superscript normal-n γ normal-→ 239 92 superscript subscript normal-U 92 238 normal-U subscript superscript 23.5 min superscript β normal-→ 239 93 Np subscript superscript 2.3565 normal-d superscript β normal-→ 239 94 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 3

U 92 238 + 0 1 n 92 239 U β - 23 min 93 239 Np β - 2.355 days 94 239 Pu subscript superscript normal-⟶ 239 92 subscript superscript 1 0 superscript subscript normal-U 92 238 normal-n normal-U subscript superscript 23 min superscript β normal-→ 239 93 Np subscript superscript 2.355 days superscript β normal-→ 239 94 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.355\ days]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 2

U 92 238 + 6 ( n , γ ) - 2 β - 94 244 Pu subscript superscript 2 superscript β 6 normal-n γ normal-→ 244 94 superscript subscript normal-U 92 238 Pu \mathrm{{}^{238}_{\ 92}U\ \xrightarrow[-2\ \beta^{-}]{+\ 6\ (n,\gamma)}\ ^{244% }_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 3

Es 99 254 + 20 48 Ca 119 302 Uue * n o a t o m s fragments superscript subscript Es 99 254 subscript superscript 48 20 Ca subscript superscript normal-→ 302 119 superscript Uue normal-→ n o a t o m s \,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue% }^{*}\to\ \ no\ atoms

NTCIR12-MathWiki-6rate: 2

U 92 238 + 28 64 Ni 120 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 subscript superscript normal-→ 302 120 subscript superscript 64 28 superscript subscript normal-U 92 238 Ni superscript Ubn normal-→ 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 \,{}^{238}_{92}\mathrm{U}+\,^{64}_{28}\mathrm{Ni}\to\,^{302}_{120}\mathrm{Ubn}% ^{*}\to\ \mathit{fission\ only}

NTCIR12-MathWiki-6rate: 0

0 U V W 0 normal-→ 0 U normal-→ V normal-→ W normal-→ 0 0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0

NTCIR12-MathWiki-6rate: 0

0 U V W 0 normal-→ 0 U normal-→ V normal-→ W normal-→ 0 0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0

NTCIR12-MathWiki-6rate: 2

U 92 238 + 15 n , 7 β - 99 253 Es subscript superscript 15 normal-n 7 superscript β normal-→ 253 99 superscript subscript normal-U 92 238 Es \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{+\ 15n,7\beta^{-}}\ ^{253}_{\ 99}Es}

NTCIR12-MathWiki-6rate: 0

n + p d + γ normal-→ n p d γ n+p\rightarrow d+\gamma

NTCIR12-MathWiki-6rate: 0

γ CMB + p Δ + n + π + . normal-→ subscript γ CMB p superscript normal-Δ normal-→ n superscript π \gamma_{\rm CMB}+p\rightarrow\Delta^{+}\rightarrow n+\pi^{+}.

NTCIR12-MathWiki-6rate: 0

ν + ν ¯ Z hadrons normal-→ ν normal-¯ ν Z normal-→ hadrons \nu+\bar{\nu}\rightarrow Z\rightarrow\,\text{hadrons}

NTCIR12-MathWiki-6rate: 0

i : A B D normal-: i normal-→ A B normal-→ D i:A\to B\to D

NTCIR12-MathWiki-6rate: 0

e - + p ν e + n normal-→ superscript e p subscript ν e n e^{-}+p\to\nu_{e}+n

NTCIR12-MathWiki-6rate: 1

Be 4 7 + e 53.22 d - Li 3 7 fragments superscript subscript Be 4 7 e superscript 53.22 normal-d normal-→ superscript subscript Li 3 7 \mathrm{{}^{7}_{4}Be}+\mathrm{e{}^{-}}\ \xrightarrow{\ \mathrm{53.22d}}\ % \mathrm{{}^{7}_{3}Li}

NTCIR12-MathWiki-6rate: 4

U 92 238 + 12 24 Mg 104 259 Rf + 3 0 1 n subscript superscript normal-→ 259 104 subscript superscript 24 12 superscript subscript normal-U 92 238 Mg Rf subscript superscript 3 1 0 normal-n \,{}^{238}_{92}\mathrm{U}\ +\,^{24}_{12}\mathrm{Mg}\to\,^{259}_{104}\mathrm{Rf% }\ +3\,^{1}_{0}\mathrm{n}

NTCIR12-MathWiki-6rate: 1

M 𝔨 * 𝔱 * . normal-→ M superscript 𝔨 normal-→ superscript 𝔱 \displaystyle{M\rightarrow\mathfrak{k}^{*}\rightarrow\mathfrak{t}^{*}.}

NTCIR12-MathWiki-6rate: 2

D 1 2 + 1 3 T 2 4 He ( 3.5 MeV ) + 0 1 n ( 14.1 MeV ) subscript superscript normal-→ 4 2 subscript superscript 3 1 superscript subscript normal-D 1 2 normal-T subscript superscript 1 0 He 3.5 MeV normal-n 14.1 MeV {}^{2}_{1}\mathrm{D}+\,^{3}_{1}\mathrm{T}\rightarrow\,^{4}_{2}\mathrm{He}\left% (3.5\,\mathrm{MeV}\right)+\,^{1}_{0}\mathrm{n}\left(14.1\,\mathrm{MeV}\right)

NTCIR12-MathWiki-6rate: 0

π + W * μ + ν μ normal-→ superscript π superscript W normal-→ superscript μ subscript ν μ \pi^{+}\to W^{*}\to\mu^{+}\nu_{\mu}

NTCIR12-MathWiki-6rate: 3

U 92 238 + 0 1 n 92 239 U β - 23 min 93 239 Np β - 2.355 days 94 239 Pu subscript superscript normal-⟶ 239 92 subscript superscript 1 0 superscript subscript normal-U 92 238 normal-n normal-U subscript superscript 23 min superscript β normal-→ 239 93 Np subscript superscript 2.355 days superscript β normal-→ 239 94 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.355\ days]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 3

U 92 235 + 0 1 n 92 236 U m 120 ns 92 236 U + γ subscript superscript normal-⟶ 236 92 subscript superscript 1 0 superscript subscript normal-U 92 235 normal-n subscript normal-U normal-m subscript superscript 120 ns absent normal-→ 236 92 normal-U γ \mathrm{{}^{235}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{236}_{\ 92}U_{m}\ % \xrightarrow[120\ ns]{}\ ^{236}_{\ 92}U\ +\ \gamma}

NTCIR12-MathWiki-6rate: 3

U 92 236 + 0 1 n 92 237 U β - 6.75 d 93 237 Np subscript superscript normal-⟶ 237 92 subscript superscript 1 0 superscript subscript normal-U 92 236 normal-n normal-U subscript superscript 6.75 normal-d superscript β normal-→ 237 93 Np \mathrm{{}^{236}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{237}_{\ 92}U\ % \xrightarrow[6.75\ d]{\beta^{-}}\ ^{237}_{\ 93}Np}

NTCIR12-MathWiki-6rate: 1

ν e + n p + e - normal-→ subscript ν e n p superscript e {{\nu}_{e}}+n\to p+e^{-}

NTCIR12-MathWiki-6rate: 0

n + p d + γ normal-→ normal-→ n p d γ \vec{n}+p\to d+\gamma

NTCIR12-MathWiki-6rate: 0

A + n B * + γ normal-→ A n superscript B γ A+n\to B^{*}+\gamma

NTCIR12-MathWiki-6rate: 2

Succinate + Q Fumarate + QH 2 normal-→ Succinate normal-Q Fumarate subscript QH 2 \rm Succinate+Q\rightarrow Fumarate+QH_{2}\!

NTCIR12-MathWiki-6rate: 0

ν e + n e - + p normal-→ subscript ν e n superscript e p \nu_{e}+n\rightarrow e^{-}+p

NTCIR12-MathWiki-6rate: 3

U 92 238 + 0 1 n 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu subscript superscript normal-⟶ 239 92 subscript superscript 1 0 superscript subscript normal-U 92 238 normal-n normal-U subscript superscript 23.5 min superscript β normal-→ 239 93 Np subscript superscript 2.3565 normal-d superscript β normal-→ 239 94 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 3

U 92 238 + 0 1 n 92 239 U β - 23.5 min 93 239 Np β - 2.3565 d 94 239 Pu subscript superscript normal-⟶ 239 92 subscript superscript 1 0 superscript subscript normal-U 92 238 normal-n normal-U subscript superscript 23.5 min superscript β normal-→ 239 93 Np subscript superscript 2.3565 normal-d superscript β normal-→ 239 94 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{% \beta^{-}}\ ^{239}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 3

U 92 238 + 1 2 D 93 238 Np + 2 0 1 n ; 93 238 Np β - 2.117 d 94 238 Pu fragments superscript subscript normal-U 92 238 subscript superscript 2 1 D subscript superscript normal-⟶ 238 93 Np subscript superscript 2 1 0 n subscript superscript normal-; 238 93 Np subscript superscript 2.117 normal-d superscript β normal-→ 238 94 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{2}_{1}D\ \longrightarrow\ ^{238}_{\ 93}Np\ +\ 2% \ ^{1}_{0}n\quad;\quad^{238}_{\ 93}Np\ \xrightarrow[2.117\ d]{\beta^{-}}\ ^{23% 8}_{\ 94}Pu}

NTCIR12-MathWiki-6rate: 2

Th 90 232 + 0 1 n 90 233 Th β - 22.3 min 91 233 Pa β - 26.967 d 92 233 U subscript superscript normal-⟶ 233 90 subscript superscript 1 0 superscript subscript Th 90 232 normal-n Th subscript superscript 22.3 min superscript β normal-→ 233 91 Pa subscript superscript 26.967 normal-d superscript β normal-→ 233 92 normal-U \mathrm{{}^{232}_{\ 90}Th\ +\ ^{1}_{0}n\ \longrightarrow\ ^{233}_{\ 90}Th\ % \xrightarrow[22.3\ min]{\beta^{-}}\ ^{233}_{\ 91}Pa\ \xrightarrow[26.967\ d]{% \beta^{-}}\ ^{233}_{\ 92}U}

NTCIR12-MathWiki-6rate: 0

U 𝑗 V 𝑔 Y U j normal-→ V g normal-→ Y U\overset{j}{\to}V\overset{g}{\to}Y

NTCIR12-MathWiki-6rate: 0

A + γ A * normal-→ A γ superscript A A+\gamma\rightarrow A^{*}

NTCIR12-MathWiki-6rate: 0

T 0 + T 0 S * + S 0 + phonons normal-→ subscript T 0 subscript T 0 superscript S subscript S 0 phonons T_{0}+T_{0}\rightarrow S^{*}+S_{0}+\,\text{phonons}

NTCIR12-MathWiki-6rate: 2

Th 90 232 + n Th 90 233 + γ β - Pa 91 233 β - U 92 233 normal-→ subscript superscript Th 232 90 normal-n subscript superscript Th 233 90 γ superscript β normal-→ subscript superscript Pa 233 91 superscript β normal-→ subscript superscript normal-U 233 92 {}_{\ 90}^{232}\mathrm{Th}+\mathrm{n}\rightarrow{}_{\ 90}^{233}\mathrm{Th}+% \gamma\ \xrightarrow{\beta^{-}}\ {}_{\ 91}^{233}\mathrm{Pa}\ \xrightarrow{% \beta^{-}}\ {}_{\ 92}^{233}\mathrm{U}

NTCIR12-MathWiki-6rate: 1

n + Th 90 232 Th 90 233 β - Pa 91 233 β - U 92 233 normal-→ normal-n subscript superscript Th 232 90 subscript superscript Th 233 90 superscript β normal-→ subscript superscript Pa 233 91 superscript β normal-→ subscript superscript normal-U 233 92 \mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}% \xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ % 92}^{233}\mathrm{U}

NTCIR12-MathWiki-6rate: 2

n + Th 90 232 Th 90 233 β - Pa 91 233 β - U 92 233 + n U 92 232 + 2 n normal-→ normal-n subscript superscript Th 232 90 subscript superscript Th 233 90 superscript β normal-→ subscript superscript Pa 233 91 superscript β normal-→ subscript superscript normal-U 233 92 normal-n normal-→ subscript superscript normal-U 232 92 2 normal-n \mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}% \xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ % 92}^{233}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{232}\mathrm{U}+2\mathrm{n}

NTCIR12-MathWiki-6rate: 1

n + Th 90 232 Th 90 231 + 2 n β - Pa 91 231 + n Pa 91 232 β - U 92 232 normal-→ normal-n subscript superscript Th 232 90 subscript superscript Th 231 90 2 normal-n superscript β normal-→ subscript superscript Pa 231 91 normal-n normal-→ subscript superscript Pa 232 91 superscript β normal-→ subscript superscript normal-U 232 92 \mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{231}\mathrm{Th}+2% \mathrm{n}\xrightarrow{\beta^{-}}{}_{\ 91}^{231}\mathrm{Pa}+\mathrm{n}% \rightarrow{}_{\ 91}^{232}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ 92}^{232}% \mathrm{U}

NTCIR12-MathWiki-6rate: 0

0 U M V 0 normal-→ 0 U normal-→ M normal-→ V normal-→ 0 0\to U\to M\to V\to 0

NTCIR12-MathWiki-6rate: 3

U 92 238 + 0 1 n 92 239 U 93 239 Np + β 94 239 Pu + β subscript superscript normal-→ 239 92 subscript superscript 1 0 superscript subscript normal-U 92 238 normal-n normal-U subscript superscript normal-→ 239 93 Np β subscript superscript normal-→ 239 94 Pu β \mathrm{{}^{238}_{\ 92}U+\,^{1}_{0}n\;\rightarrow\;^{239}_{\ 92}U\;\rightarrow% \;^{239}_{\ 93}Np+\beta\;\rightarrow\;^{239}_{\ 94}Pu+\beta}

NTCIR12-MathWiki-6rate: 4

U 92 238 + 28 64 Ni 120 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 subscript superscript normal-→ 302 120 subscript superscript 64 28 superscript subscript normal-U 92 238 Ni superscript Ubn normal-→ 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 \,{}^{238}_{92}\mathrm{U}+\,^{64}_{28}\mathrm{Ni}\to\,^{302}_{120}\mathrm{Ubn}% ^{*}\to\ \mathit{fission\ only}

NTCIR12-MathWiki-6rate: 3

Es 99 254 + 20 48 Ca 119 302 Uue * subscript superscript normal-→ 302 119 subscript superscript 48 20 superscript subscript Es 99 254 Ca superscript Uue \,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue% }^{*}

NTCIR12-MathWiki-6rate: 4

U 92 238 + n U 92 239 + γ β - Np 93 239 β - Pu 94 239 normal-→ subscript superscript normal-U 238 92 normal-n subscript superscript normal-U 239 92 γ superscript β normal-→ subscript superscript Np 239 93 superscript β normal-→ subscript superscript Pu 239 94 {}_{\ 92}^{238}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{239}\mathrm{U}+% \gamma\xrightarrow{\beta^{-}}{}_{\ 93}^{239}\mathrm{Np}\xrightarrow{\beta^{-}}% {}_{\ 94}^{239}\mathrm{Pu}

NTCIR12-MathWiki-6rate: 1

U + W U W U+W

NTCIR12-MathWiki-6rate: 0

U + W U W U+W

NTCIR12-MathWiki-6rate: 0

U + W U W U+W

NTCIR12-MathWiki-6rate: 0

U + W U W U+W

NTCIR12-MathWiki-7rate: 2

0 M M M ′′ 0 normal-→ 0 superscript M normal-′ normal-→ M normal-→ superscript M ′′ normal-→ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 V V V ′′ 0 , normal-→ 0 superscript V normal-′ normal-→ V normal-→ superscript V ′′ normal-→ 0 0\to V^{\prime}\to V\to V^{\prime\prime}\to 0,

NTCIR12-MathWiki-7rate: 2

0 V P T P d π π * T M 0 normal-→ 0 V P normal-→ T P d π normal-→ superscript π T M normal-→ 0 0\to VP\to TP\xrightarrow{d\pi}\pi^{*}TM\to 0

NTCIR12-MathWiki-7rate: 2

0 E E E ′′ 0 normal-→ 0 superscript E normal-′ normal-→ E normal-→ superscript E ′′ normal-→ 0 \ 0\to E^{\prime}\to E\to E^{\prime\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 ker h H n ( C ; G ) h Hom ( H n ( C ) , G ) 0. normal-→ 0 kernel h normal-→ superscript H n C G superscript normal-→ h Hom subscript H n C G normal-→ 0. 0\rightarrow\ker h\rightarrow H^{n}(C;G)\stackrel{h}{\rightarrow}\,\text{Hom}(% H_{n}(C),G)\rightarrow 0.

NTCIR12-MathWiki-7rate: 2

0 𝒪 𝒪 * 0 normal-→ 0 normal-→ 𝒪 normal-→ superscript 𝒪 normal-→ 0 0\to\mathbb{Z}\to\mathcal{O}\to\mathcal{O}^{*}\to 0

NTCIR12-MathWiki-7rate: 2

0 2 π i 𝐎 exp 𝐎 * 0 normal-→ 0 2 π i normal-→ 𝐎 normal-→ superscript 𝐎 normal-→ 0 0\to 2\pi i\mathbb{Z}\to\mathbf{O}\xrightarrow{\exp}\mathbf{O}^{*}\to 0

NTCIR12-MathWiki-7rate: 2

0 M M M ′′ 0 normal-→ 0 superscript M normal-′ normal-→ M normal-→ superscript M ′′ normal-→ 0 0\rightarrow M^{\prime}\rightarrow M\rightarrow M^{\prime\prime}\rightarrow 0

NTCIR12-MathWiki-7rate: 1

X - 1 d - 1 X 0 d 0 X 1 d 1 X 2 normal-→ normal-⋯ superscript X 1 superscript d 1 normal-→ superscript X 0 superscript d 0 normal-→ superscript X 1 superscript d 1 normal-→ superscript X 2 normal-→ normal-⋯ \cdots\to X^{-1}\xrightarrow{d^{-1}}X^{0}\xrightarrow{d^{0}}X^{1}\xrightarrow{% d^{1}}X^{2}\to\cdots

NTCIR12-MathWiki-7rate: 2

0 X Y Z 0 normal-→ 0 X normal-→ Y normal-→ Z normal-→ 0 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0

NTCIR12-MathWiki-7rate: 3

0 H i G π G / H 0 normal-→ 0 H superscript normal-→ i G superscript normal-→ π G H normal-→ 0 0\to H\stackrel{i}{{}\to{}}G\stackrel{\pi}{{}\to{}}G/H\to 0

NTCIR12-MathWiki-7rate: 2

0 G p + 1 G p K p , * 0 normal-→ 0 superscript G p 1 normal-→ superscript G p normal-→ superscript K p normal-→ 0 0\to G^{p+1}\to G^{p}\to K^{p,*}\to 0

NTCIR12-MathWiki-7rate: 2

0 M 𝑓 M M ′′ 0 , normal-→ 0 superscript M normal-′ f normal-→ M normal-→ superscript M ′′ normal-→ 0 0\to M^{\prime}\xrightarrow{f}M\to M^{\prime\prime}\to 0,

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0

NTCIR12-MathWiki-7rate: 4

0 G π X ı H 0 normal-→ 0 superscript G superscript normal-→ superscript π superscript X superscript normal-→ superscript ı superscript H normal-→ 0 0\to G^{\wedge}\stackrel{\pi^{\wedge}}{\to}X^{\wedge}\stackrel{\imath^{\wedge}% }{\to}H^{\wedge}\to 0

NTCIR12-MathWiki-7rate: 2

0 U V W 0 normal-→ 0 U normal-→ V normal-→ W normal-→ 0 0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 U V W 0 normal-→ 0 U normal-→ V normal-→ W normal-→ 0 0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 B E A 0 normal-→ 0 B normal-→ E normal-→ A normal-→ 0 0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 B E A 0 normal-→ 0 B normal-→ E normal-→ A normal-→ 0 0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 B Y A 0 normal-→ 0 B normal-→ Y normal-→ A normal-→ 0 0\rightarrow B\rightarrow Y\rightarrow A\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 B E A 0 normal-→ 0 B normal-→ E normal-→ A normal-→ 0 0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 B E A 0 normal-→ 0 B normal-→ superscript E normal-′ normal-→ A normal-→ 0 0\rightarrow B\rightarrow E^{\prime}\rightarrow A\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 I L L 0 normal-→ 0 I normal-→ L normal-→ superscript L normal-′ normal-→ 0 0\to I\to L\to L^{\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0

NTCIR12-MathWiki-7rate: 2

0 L M N 0 normal-→ 0 L normal-→ M normal-→ N normal-→ 0 0\to L\to M\to N\to 0

NTCIR12-MathWiki-7rate: 2

0 𝔞 𝔢 𝔤 0 normal-→ 0 𝔞 normal-→ 𝔢 normal-→ 𝔤 normal-→ 0 0\rightarrow\mathfrak{a}\rightarrow\mathfrak{e}\rightarrow\mathfrak{g}\rightarrow 0

NTCIR12-MathWiki-7rate: 2

1 K i G π H 1 normal-→ 1 K superscript normal-→ i G superscript normal-→ π H normal-→ 1 1\rightarrow K\stackrel{i}{\rightarrow}G\stackrel{\pi}{\rightarrow}H\rightarrow 1

NTCIR12-MathWiki-7rate: 2

1 K i G π H 1 normal-→ 1 K superscript normal-→ superscript i normal-′ superscript G normal-′ superscript normal-→ superscript π normal-′ H normal-→ 1 1\to K\stackrel{i^{\prime}}{\rightarrow}G^{\prime}\stackrel{\pi^{\prime}}{% \rightarrow}H\rightarrow 1

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\;\rightarrow\;A\;\rightarrow\;B\;\rightarrow\;C\;\rightarrow\;0

NTCIR12-MathWiki-7rate: 1

0 C 0 d 0 C 1 d 1 C 2 d 2 d n - 1 C n 0. normal-→ 0 superscript C 0 superscript normal-⟶ subscript d 0 superscript C 1 superscript normal-⟶ subscript d 1 superscript C 2 superscript normal-⟶ subscript d 2 normal-⋯ superscript normal-⟶ subscript d n 1 superscript C n normal-⟶ 0. 0\to C^{0}\stackrel{d_{0}}{\longrightarrow}C^{1}\stackrel{d_{1}}{% \longrightarrow}C^{2}\stackrel{d_{2}}{\longrightarrow}\cdots\stackrel{d_{n-1}}% {\longrightarrow}C^{n}\longrightarrow 0.

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0

NTCIR12-MathWiki-7rate: 2

0 X V 0. normal-→ 0 normal-→ X normal-→ V normal-→ 0. 0\rightarrow\mathbb{R}\rightarrow X\rightarrow V\rightarrow 0.\,\!

NTCIR12-MathWiki-7rate: 2

0 R d 2 R 2 d 1 R 0 , normal-→ 0 R subscript d 2 normal-→ superscript R 2 subscript d 1 normal-→ R normal-→ 0 0\to R\xrightarrow{\ d_{2}\ }R^{2}\xrightarrow{\ d_{1}\ }R\to 0,

NTCIR12-MathWiki-7rate: 2

0 L M N 0 normal-→ 0 L normal-→ M normal-→ N normal-→ 0 0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 M 𝔥 𝔤 0 normal-→ 0 M normal-→ 𝔥 normal-→ 𝔤 normal-→ 0 0\rightarrow M\rightarrow\mathfrak{h}\rightarrow\mathfrak{g}\rightarrow 0

NTCIR12-MathWiki-7rate: 2

0 A A A ′′ 0 normal-→ 0 superscript A normal-′ normal-→ A normal-→ superscript A ′′ normal-→ 0 0\rightarrow A^{\prime}\rightarrow A\rightarrow A^{\prime\prime}\rightarrow 0

NTCIR12-MathWiki-7rate: 0

G superscript G G^{\wedge}

NTCIR12-MathWiki-7rate: 0

π G π superscript G \pi\in G^{\wedge}

NTCIR12-MathWiki-7rate: 2

0 A B P 0 normal-→ 0 A normal-→ B normal-→ P normal-→ 0 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0\,

NTCIR12-MathWiki-7rate: 2

0 M M M ′′ 0 normal-→ 0 superscript M normal-′ normal-→ M normal-→ superscript M ′′ normal-→ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0\;

NTCIR12-MathWiki-7rate: 2

0 R r - s X Y 0 normal-→ 0 R r s normal-→ X normal-→ Y normal-→ 0 0\to R\xrightarrow{r-s}X\to Y\to 0

NTCIR12-MathWiki-7rate: 0

n A superscript n A n^{\wedge}A

NTCIR12-MathWiki-7rate: 2

0 M M M ′′ 0 normal-→ 0 superscript M normal-′ normal-→ M normal-→ superscript M ′′ normal-→ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 K X P 0 , normal-→ 0 K normal-→ X normal-→ superscript P normal-′ normal-→ 0 0\rightarrow K\rightarrow X\rightarrow P^{\prime}\rightarrow 0,

NTCIR12-MathWiki-7rate: 2

0 M M M ′′ 0 normal-→ 0 superscript M normal-′ normal-→ M normal-→ superscript M ′′ normal-→ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 Pic ( X ) δ Br ( K ) Br ( K ) ( X ) 0 . normal-→ 0 Pic X normal-→ superscript normal-→ δ Br K normal-→ Br K X normal-→ 0 . 0\rightarrow\mathrm{Pic}(X)\rightarrow\mathbb{Z}\stackrel{\delta}{\rightarrow}% \mathrm{Br}(K)\rightarrow\mathrm{Br}(K)(X)\rightarrow 0\ .

NTCIR12-MathWiki-7rate: 2

0 𝒜 ϕ ψ 𝒞 0 normal-→ 0 𝒜 superscript normal-→ ϕ superscript normal-→ ψ 𝒞 normal-→ 0 0\ \rightarrow\mathcal{A}\ \stackrel{\phi}{\rightarrow}\ \mathcal{B}\ % \stackrel{\psi}{\rightarrow}\ \mathcal{C}\ \rightarrow\ 0

NTCIR12-MathWiki-7rate: 2

0 F G H 0. normal-→ 0 F normal-→ G normal-→ H normal-→ 0. 0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.

NTCIR12-MathWiki-7rate: 3

0 A q B r C 0 normal-→ 0 A superscript normal-⟶ q B superscript normal-⟶ r C normal-→ 0 0\rightarrow A\stackrel{q}{\longrightarrow}B\stackrel{r}{\longrightarrow}C% \rightarrow 0\,

NTCIR12-MathWiki-7rate: 2

0 X E Y 0 normal-→ 0 X normal-→ E normal-→ Y normal-→ 0 0\to X\to E\to Y\to 0\,

NTCIR12-MathWiki-7rate: 2

0 M M M ′′ 0 normal-→ 0 superscript M normal-′ normal-→ M normal-→ superscript M ′′ normal-→ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 M M M ′′ 0 normal-→ 0 superscript M normal-′ normal-→ M normal-→ superscript M ′′ normal-→ 0 0\to M^{\prime}\to M\to M^{\prime\prime}\to 0

NTCIR12-MathWiki-7rate: 2

0 U M V 0 normal-→ 0 U normal-→ M normal-→ V normal-→ 0 0\to U\to M\to V\to 0

NTCIR12-MathWiki-7rate: 3

W f X g Y h Z superscript normal-→ f W X superscript normal-→ g Y superscript normal-→ h Z W\stackrel{f}{\ \to\ }X\stackrel{g}{\ \to\ }Y\stackrel{h}{\ \to\ }Z

NTCIR12-MathWiki-7rate: 2

0 A B C 0 normal-→ 0 A normal-→ B normal-→ C normal-→ 0 0\to A\to B\to C\to 0

NTCIR12-MathWiki-8rate: 3

| a | = { a , if a 0 - a , if a 0 a cases a if a 0 a if a 0 |a|=\begin{cases}a,&\mbox{if }~{}a\geq 0\\ -a,&\mbox{if }~{}a\leq 0\end{cases}\;

NTCIR12-MathWiki-8rate: 0

λ ( n ) = { ϕ ( n ) if n = 2 , 3 , 4 , 5 , 7 , 9 , 11 , 13 , 17 , 19 , 23 , 25 , 27 , 1 2 ϕ ( n ) if n = 8 , 16 , 32 , 64 , λ n cases ϕ n if n 2 3 4 5 7 9 11 13 17 19 23 25 27 normal-… 1 2 ϕ n if n 8 16 32 64 normal-… \lambda(n)=\begin{cases}\;\;\phi(n)&\,\text{if }n=2,3,4,5,7,9,11,13,17,19,23,2% 5,27,\dots\\ \tfrac{1}{2}\phi(n)&\,\text{if }n=8,16,32,64,\dots\end{cases}

NTCIR12-MathWiki-8rate: 2

f ( x ) = { 1 x if x > 0 , 5 if x 0. f x cases 1 x if x 0 5 if x 0. f(x)=\begin{cases}\frac{1}{x}&\mbox{if }~{}x>0,\\ 5&\mbox{if }~{}x\leq 0.\end{cases}

NTCIR12-MathWiki-8rate: 0

a ( n ) = { 2 n + 1 - 2 3 , when n is even, 2 n + 1 - 1 3 , when n is odd. a n cases superscript 2 n 1 2 3 when n is even, superscript 2 n 1 1 3 when n is odd. a(n)=\begin{cases}\frac{2^{n+1}-2}{3},&\,\text{when }n\,\text{ is even,}\\ \frac{2^{n+1}-1}{3},&\,\text{when }n\,\text{ is odd.}\end{cases}

NTCIR12-MathWiki-8rate: 1

log * ( n ) = { 0 , if n 1 1 + log * ( log n ) , if n > 1 superscript n cases 0 if n 1 1 superscript n if n 1 \log^{*}(n)=\begin{cases}0,&\,\text{if }n\leq 1\\ 1+\log^{*}(\log n),&\,\text{if }n>1\end{cases}

NTCIR12-MathWiki-8rate: 2

( n k ) = { n k ¯ / k ! if k n 2 n n - k ¯ / ( n - k ) ! if k > n 2 . binomial n k cases superscript n normal-¯ k k if k n 2 superscript n normal-¯ n k n k if k n 2 {\left({{n}\atop{k}}\right)}=\begin{cases}n^{\underline{k}}/k!&\,\text{if }\ k% \leq\frac{n}{2}\\ n^{\underline{n-k}}/(n-k)!&\,\text{if }\ k>\frac{n}{2}\end{cases}.

NTCIR12-MathWiki-8rate: 4

Y = { 0 , if Y * > 0 1 , if Y * < 0. Y cases 0 if superscript Y 0 1 if superscript Y 0. Y=\begin{cases}0,&\mbox{if }~{}Y^{*}>0\\ 1,&\mbox{if }~{}Y^{*}<0.\end{cases}

NTCIR12-MathWiki-8rate: 2

Y n = { 1 , if U n > 0 , 0 , if U n 0 subscript Y n cases 1 if subscript U n 0 0 if subscript U n 0 Y_{n}=\begin{cases}1,&\,\text{if }U_{n}>0,\\ 0,&\,\text{if }U_{n}\leq 0\end{cases}

NTCIR12-MathWiki-8rate: 2

Y n = { 1 , if U n > 0 , 0 , if U n 0 subscript Y n cases 1 if subscript U n 0 0 if subscript U n 0 Y_{n}=\begin{cases}1,&\,\text{if }U_{n}>0,\\ 0,&\,\text{if }U_{n}\leq 0\end{cases}

NTCIR12-MathWiki-8rate: 0

λ ( n ) = { φ ( n ) if n = 2 , 3 , 4 , 5 , 6 , 7 , 9 , 10 , 11 , 13 , 14 , 17 , 19 , 22 , 23 , 25 , 26 , 27 , 29 1 2 φ ( n ) if n = 8 , 16 , 32 , 64 , 128 , 256 λ n cases φ n if n 2 3 4 5 6 7 9 10 11 13 14 17 19 22 23 25 26 27 29 normal-… 1 2 φ n if n 8 16 32 64 128 256 normal-… \lambda(n)=\begin{cases}\;\;\varphi(n)&\mbox{if }~{}n=2,3,4,5,6,7,9,10,11,13,1% 4,17,19,22,23,25,26,27,29\dots\\ \tfrac{1}{2}\varphi(n)&\,\text{if }n=8,16,32,64,128,256\dots\end{cases}

NTCIR12-MathWiki-8rate: 1

T j ( x ) U k ( x ) = { 1 2 ( U j + k ( x ) + U k - j ( x ) ) , if k j - 1. 1 2 ( U j + k ( x ) - U j - k - 2 ( x ) ) , if k j - 2. subscript T j x subscript U k x cases 1 2 subscript U j k x subscript U k j x if k j 1. 1 2 subscript U j k x subscript U j k 2 x if k j 2. T_{j}(x)U_{k}(x)=\begin{cases}\tfrac{1}{2}\left(U_{j+k}(x)+U_{k-j}(x)\right),&% \,\text{if }k\geq j-1.\\ \tfrac{1}{2}\left(U_{j+k}(x)-U_{j-k-2}(x)\right),&\,\text{if }k\leq j-2.\end{cases}

NTCIR12-MathWiki-8rate: 1

V out = { V S + if V 1 > V 2 , V S - if V 1 < V 2 , 0 if V 1 = V 2 , subscript V out cases subscript V limit-from S if subscript V 1 subscript V 2 subscript V limit-from S if subscript V 1 subscript V 2 0 if subscript V 1 subscript V 2 V_{\,\text{out}}=\begin{cases}V_{\,\text{S}+}&\,\text{if }V_{1}>V_{2},\\ V_{\,\text{S}-}&\,\text{if }V_{1}<V_{2},\\ 0&\,\text{if }V_{1}=V_{2},\end{cases}

NTCIR12-MathWiki-8rate: 0

P * = 1 2 superscript P 1 2 P^{*}=\frac{1}{2}

NTCIR12-MathWiki-8rate: 1

{ x * log ( x * ) - x * if x * > 0 0 if x * = 0 cases superscript x superscript x superscript x if superscript x 0 0 if superscript x 0 \begin{cases}x^{*}\log(x^{*})-x^{*}&\,\text{if }x^{*}>0\\ 0&\,\text{if }x^{*}=0\end{cases}

NTCIR12-MathWiki-8rate: 2

f ( x * ) = { 0 , | x * | 1 , | x * | > 1. superscript f normal-⋆ superscript x cases 0 superscript x 1 superscript x 1. f^{\star}\left(x^{*}\right)=\begin{cases}0,&\left|x^{*}\right|\leq 1\\ \infty,&\left|x^{*}\right|>1.\end{cases}

NTCIR12-MathWiki-8rate: 1

v i , j = { n - 1 2 j = 1 2 n cos ( π ( j - 1 ) ( i - 1 2 ) n ) o t h e r w i s e subscript v i j cases superscript n 1 2 j 1 2 n π j 1 i 1 2 n o t h e r w i s e v_{i,j}=\begin{cases}n^{-\frac{1}{2}}&j=1\\ \sqrt{\frac{2}{n}}\cos(\frac{\pi(j-1)(i-\frac{1}{2})}{n})&otherwise\end{cases}

NTCIR12-MathWiki-8rate: 1

ζ = { 1 if n 0 1 / 2 if n = 0 , ξ = { 1 if m 0 1 / 2 if m = 0 formulae-sequence ζ cases 1 if n 0 1 2 if n 0 ξ cases 1 if m 0 1 2 if m 0 \zeta=\begin{cases}1&\mbox{if }~{}n\neq 0\\ 1/2&\mbox{if }~{}n=0\end{cases},\quad\xi=\begin{cases}1&\mbox{if }~{}m\neq 0\\ 1/2&\mbox{if }~{}m=0\end{cases}

NTCIR12-MathWiki-8rate: 1

c n = def { A n 2 i e i ϕ n = 1 2 ( a n - i b n ) for n > 0 1 2 a 0 for n = 0 c | n | * for n < 0. superscript def subscript c n cases subscript A n 2 i superscript e i subscript ϕ n 1 2 subscript a n i subscript b n for n 0 1 2 subscript a 0 for n 0 superscript subscript c n for n 0. c_{n}\ \stackrel{\mathrm{def}}{=}\ \begin{cases}\frac{A_{n}}{2i}e^{i\phi_{n}}=% \frac{1}{2}(a_{n}-ib_{n})&\,\text{for }n>0\\ \frac{1}{2}a_{0}&\,\text{for }n=0\\ c_{|n|}^{*}&\,\text{for }n<0.\end{cases}

NTCIR12-MathWiki-8rate: 1

ψ ( t ) = { 1 0 t < 1 2 , - 1 1 2 t < 1 , 0 otherwise. ψ t cases 1 0 t 1 2 1 1 2 t 1 0 otherwise. \psi(t)=\begin{cases}1&0\leq t<\frac{1}{2},\\ -1&\frac{1}{2}\leq t<1,\\ 0&\mbox{otherwise.}\end{cases}

NTCIR12-MathWiki-8rate: 1

H [ n ] = { 0 , n < 0 , 1 / 2 , n = 0 , 1 , n > 0 , H delimited-[] n cases 0 n 0 1 2 n 0 1 n 0 H[n]=\begin{cases}0,&n<0,\\ 1/2,&n=0,\\ 1,&n>0,\end{cases}

NTCIR12-MathWiki-8rate: 0

f ( x ) = { 1 - 1 x < 0 1 2 x = 0 1 - x 2 otherwise f x cases 1 1 x 0 1 2 x 0 1 superscript x 2 otherwise f(x)=\begin{cases}1&-1\leq x<0\\ \frac{1}{2}&x=0\\ 1-x^{2}&\,\text{otherwise}\end{cases}

NTCIR12-MathWiki-8rate: 2

x + α = { x α if x > 0 0 otherwise superscript subscript x α cases superscript x α if x 0 0 otherwise x_{+}^{\alpha}=\begin{cases}x^{\alpha}&\,\text{if }x>0\\ 0&\,\text{otherwise}\end{cases}

NTCIR12-MathWiki-8rate: 2

log * n := { 0 if n 1 ; 1 + log * ( log n ) if n > 1 assign superscript n cases 0 if n 1 1 superscript n if n 1 \log^{*}n:=\begin{cases}0&\mbox{if }~{}n\leq 1;\\ 1+\log^{*}(\log n)&\mbox{if }~{}n>1\end{cases}

NTCIR12-MathWiki-8rate: 1

δ i j = { 0 if i j , 1 if i = j . subscript δ i j cases 0 if i j 1 if i j \delta_{ij}=\begin{cases}0&\,\text{if }i\neq j,\\ 1&\,\text{if }i=j.\end{cases}

NTCIR12-MathWiki-8rate: 1

f ( t ) = { t 1 / 3 if t > ( 6 29 ) 3 1 3 ( 29 6 ) 2 t + 4 29 otherwise f t cases superscript t 1 3 if t superscript 6 29 3 1 3 superscript 29 6 2 t 4 29 otherwise f(t)=\begin{cases}t^{1/3}&\,\text{if }t>(\frac{6}{29})^{3}\\ \frac{1}{3}\left(\frac{29}{6}\right)^{2}t+\frac{4}{29}&\,\text{otherwise}\end{cases}

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f - 1 ( t ) = { t 3 if t > 6 29 3 ( 6 29 ) 2 ( t - 4 29 ) otherwise superscript f 1 t cases superscript t 3 if t 6 29 3 superscript 6 29 2 t 4 29 otherwise f^{-1}(t)=\begin{cases}t^{3}&\,\text{if }t>\tfrac{6}{29}\\ 3\left(\tfrac{6}{29}\right)^{2}\left(t-\tfrac{4}{29}\right)&\,\text{otherwise}% \end{cases}

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f * ( x * ) = { 2 x * - 4 , x * < 4 x * 2 4 , 4 x * 6 , 3 x * - 9 , x * > 6 superscript f superscript x cases 2 superscript x 4 superscript x 4 superscript superscript x 2 4 4 superscript x 6 3 superscript x 9 superscript x 6 f^{*}(x^{*})=\begin{cases}2x^{*}-4,&x^{*}<4\\ \frac{{x^{*}}^{2}}{4},&4\leqslant x^{*}\leqslant 6,\\ 3x^{*}-9,&x^{*}>6\end{cases}

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absolute margin of victory = { 0 ; w c 2 w - max { r , c 2 } ; w > c 2 absolute margin of victory cases 0 w c 2 w r c 2 w c 2 \mbox{absolute margin of victory}~{}=\begin{cases}0;&w\leq\frac{c}{2}\\ w-\max\{r,\frac{c}{2}\};&w>\frac{c}{2}\end{cases}

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Y i = { 1 if Y i 1 > Y i 0 , 0 otherwise. subscript Y i cases 1 if superscript subscript Y i 1 normal-∗ superscript subscript Y i 0 normal-∗ 0 otherwise. Y_{i}=\begin{cases}1&\,\text{if }Y_{i}^{1\ast}>Y_{i}^{0\ast},\\ 0&\,\text{otherwise.}\end{cases}

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δ b a = { 1 if a = b , 0 if a b . subscript superscript δ a b cases 1 if a b 0 if a b \delta^{a}_{b}=\begin{cases}1&\mbox{if }~{}a=b,\\ 0&\mbox{if }~{}a\neq b.\end{cases}

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Π ( t ) = def { 1 if | t | < 1 2 , 0 if | t | > 1 2 . superscript def normal-Π t cases 1 if t 1 2 0 if t 1 2 \Pi(t)\ \stackrel{\,\text{def}}{=}\ \begin{cases}1&\,\text{if }|t|<\frac{1}{2}% ,\\ 0&\,\text{if }|t|>\frac{1}{2}.\end{cases}

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M ( n ) = { n - 10 , if n > 100 M ( M ( n + 11 ) ) , if n 100 fragments M fragments normal-( n normal-) fragments normal-{ n 10 if n 100 M M n 11 if n 100 M(n)=\left\{\begin{matrix}n-10,&\mbox{if }~{}n>100\mbox{ }\\ M(M(n+11)),&\mbox{if }~{}n\leq 100\mbox{ }\end{matrix}\right.

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f ( x ) = { 1 , if | x | < 1 2 , 0 , otherwise . f x cases 1 if x 1 2 0 otherwise f(x)=\begin{cases}1,&\mathrm{if}~{}|x|<\tfrac{1}{2},\\ 0,&\mathrm{otherwise.}\end{cases}

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P i j ( ν ) = { 1 4 + 3 4 e - 4 ν / 3 if i = j 1 4 - 1 4 e - 4 ν / 3 if i j subscript P i j ν cases 1 4 3 4 superscript e 4 ν 3 if i j 1 4 1 4 superscript e 4 ν 3 if i j P_{ij}(\nu)=\left\{\begin{array}[]{cc}{1\over 4}+{3\over 4}e^{-4\nu/3}&\mbox{ % if }~{}i=j\\ {1\over 4}-{1\over 4}e^{-4\nu/3}&\mbox{ if }~{}i\neq j\end{array}\right.

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w = { w m i n , if w m i n w ( L ) w ( L ) , if w m i n w ( L ) w cases subscript w m i n if subscript w m i n w L w L if subscript w m i n w L w=\begin{cases}w_{min},&\mbox{if }~{}w_{min}\geq\;w(L)\\ w(L),&\mbox{if }~{}w_{min}\leq\;w(L)\end{cases}\,\!

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𝟏 A ( x ) = { 1 if x A , 0 if x A . subscript 1 A x cases 1 if x A 0 if x A \mathbf{1}_{A}(x)=\begin{cases}1&\,\text{if }x\in A,\\ 0&\,\text{if }x\notin A.\end{cases}

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Y 1 = { 1 if Y 1 * > 0 , 0 otherwise , subscript Y 1 cases 1 if subscript superscript Y 1 0 0 otherwise Y_{1}=\begin{cases}1&\,\text{if }Y^{*}_{1}>0,\\ 0&\,\text{otherwise},\end{cases}

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Y 2 = { 1 if Y 2 * > 0 , 0 otherwise , subscript Y 2 cases 1 if subscript superscript Y 2 0 0 otherwise Y_{2}=\begin{cases}1&\,\text{if }Y^{*}_{2}>0,\\ 0&\,\text{otherwise},\end{cases}

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f ( x ) = { exp ( - 1 / x ) if x > 0 , 0 if x 0 , f x cases 1 x if x 0 0 if x 0 f(x)=\begin{cases}\exp(-1/x)&\,\text{if }x>0,\\ 0&\,\text{if }x\leq 0,\end{cases}

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w = { w * if w * > 1 2 , 1 2 if w * 1 2 . w cases superscript w if superscript w 1 2 1 2 if superscript w 1 2 w=\begin{cases}w^{*}&\mbox{if }~{}w^{*}>\frac{1}{2},\\ \frac{1}{2}&\mbox{if }~{}w^{*}\leq\frac{1}{2}.\\ \end{cases}

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A * A = ( A * A ) 1 2 ( A * A ) 1 2 , superscript A A superscript superscript A A 1 2 superscript superscript A A 1 2 A^{*}A=(A^{*}A)^{\frac{1}{2}}(A^{*}A)^{\frac{1}{2}},

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y = { 0 if y * μ 1 , 1 if μ 1 < y * μ 2 , 2 if μ 2 < y * μ 3 , N if μ N < y * . y cases 0 if superscript y subscript μ 1 1 if subscript μ 1 superscript y subscript μ 2 2 if subscript μ 2 superscript y subscript μ 3 normal-⋮ otherwise N if subscript μ N superscript y y=\begin{cases}0&\,\text{if }y^{*}\leq\mu_{1},\\ 1&\,\text{if }\mu_{1}<y^{*}\leq\mu_{2},\\ 2&\,\text{if }\mu_{2}<y^{*}\leq\mu_{3},\\ \vdots\\ N&\,\text{if }\mu_{N}<y^{*}.\end{cases}

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y = { 0 if y * 0 , 1 if 0 < y * μ 1 , 2 if μ 1 < y * μ 2 N if μ N - 1 < y * . y cases 0 if superscript y 0 otherwise 1 if 0 superscript y subscript μ 1 otherwise 2 if subscript μ 1 superscript y subscript μ 2 otherwise normal-⋮ otherwise N if subscript μ N 1 superscript y otherwise y=\begin{cases}0~{}~{}\,\text{if}~{}~{}y^{*}\leq 0,\\ 1~{}~{}\,\text{if}~{}~{}0<y^{*}\leq\mu_{1},\\ 2~{}~{}\,\text{if}~{}~{}\mu_{1}<y^{*}\leq\mu_{2}\\ \vdots\\ N~{}~{}\,\text{if}~{}~{}\mu_{N-1}<y^{*}.\end{cases}

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y = { 1 if y * θ 1 , 2 if θ 1 < y * θ 2 , 3 if θ 2 < y * θ 3 K if θ K - 1 < y * . y cases 1 if superscript y subscript θ 1 otherwise 2 if subscript θ 1 superscript y subscript θ 2 otherwise 3 if subscript θ 2 superscript y subscript θ 3 otherwise normal-⋮ otherwise K if subscript θ K 1 superscript y otherwise y=\begin{cases}1~{}~{}\,\text{if}~{}~{}y^{*}\leq\theta_{1},\\ 2~{}~{}\,\text{if}~{}~{}\theta_{1}<y^{*}\leq\theta_{2},\\ 3~{}~{}\,\text{if}~{}~{}\theta_{2}<y^{*}\leq\theta_{3}\\ \vdots\\ K~{}~{}\,\text{if}~{}~{}\theta_{K-1}<y^{*}.\end{cases}

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A * A = ( A * A ) 1 2 ( A * A ) 1 2 , superscript A A superscript superscript A A 1 2 superscript superscript A A 1 2 A^{*}A=(A^{*}A)^{\frac{1}{2}}(A^{*}A)^{\frac{1}{2}},

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ω = { D if D 2 , 3 ( mod 4 ) 1 + D 2 if D 1 ( mod 4 ) ω cases D if D 2 annotated 3 pmod 4 1 D 2 if D annotated 1 pmod 4 \omega=\begin{cases}\sqrt{D}&\mbox{if }~{}D\equiv 2,3\;\;(\mathop{{\rm mod}}4)% \\ {{1+\sqrt{D}}\over 2}&\mbox{if }~{}D\equiv 1\;\;(\mathop{{\rm mod}}4)\end{cases}

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p q = 1 2 ( p * q + q * p ) = 1 2 ( p q * + q p * ) . normal-⋅ p q 1 2 superscript p q superscript q p 1 2 p superscript q q superscript p p\cdot q=\textstyle\frac{1}{2}(p^{*}q+q^{*}p)=\textstyle\frac{1}{2}(pq^{*}+qp^% {*}).

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rect ( t ) = Π ( t ) = { 0 if | t | > 1 2 1 2 if | t | = 1 2 1 if | t | < 1 2 . rect t normal-Π t cases 0 if t 1 2 1 2 if t 1 2 1 if t 1 2 \mathrm{rect}(t)=\Pi(t)=\begin{cases}0&\mbox{if }~{}|t|>\frac{1}{2}\\ \frac{1}{2}&\mbox{if }~{}|t|=\frac{1}{2}\\ 1&\mbox{if }~{}|t|<\frac{1}{2}.\\ \end{cases}

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F ^ ( x ) = { Φ ( w ^ ) + ϕ ( w ^ ) ( 1 w ^ - 1 u ^ ) for x μ 1 2 + K ′′′ ( 0 ) 6 2 π K ′′ ( 0 ) 3 / 2 for x = μ normal-^ F x cases normal-Φ normal-^ w ϕ normal-^ w 1 normal-^ w 1 normal-^ u for x μ 1 2 superscript K ′′′ 0 6 2 π superscript K ′′ superscript 0 3 2 for x μ \hat{F}(x)=\begin{cases}\Phi(\hat{w})+\phi(\hat{w})(\frac{1}{\hat{w}}-\frac{1}% {\hat{u}})&\,\text{for }x\neq\mu\\ \frac{1}{2}+\frac{K^{\prime\prime\prime}(0)}{6\sqrt{2\pi}K^{\prime\prime}(0)^{% 3/2}}&\,\text{for }x=\mu\end{cases}

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𝐚 i = { 1 if w i > 0 , 0 if w i < 0. subscript 𝐚 i cases 1 if subscript w i 0 0 if subscript w i 0. \mathbf{a}_{i}=\begin{cases}1&\,\text{if }w_{i}>0,\\ 0&\,\text{if }w_{i}<0.\end{cases}

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l l * = 1 2 l superscript l 1 2 {\mathit{l}\over\mathit{l}^{*}}={1\over 2}

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χ A ( x ) = { 1 if x A , 0 if x A . subscript χ A x cases 1 if x A 0 if x A \chi_{A}(x)=\begin{cases}1&\mbox{if }~{}x\in A,\\ 0&\mbox{if }~{}x\notin A.\\ \end{cases}

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f img = { f + 2 f IF , if f LO > f (high side injection) f - 2 f IF , if f LO < f (low side injection) subscript f img cases f 2 subscript f IF if subscript f LO f (high side injection) f 2 subscript f IF if subscript f LO f (low side injection) f_{\mathrm{img}}=\begin{cases}f+2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}% >f\mbox{ (high side injection)}\\ f-2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}<f\mbox{ (low side injection)}% \end{cases}

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f : ; f ( x ) = { e - 1 x 2 x > 0 , 0 x 0. normal-: f formulae-sequence normal-→ f x cases superscript e 1 superscript x 2 x 0 0 x 0. f:\mathbb{R}\to\mathbb{R};\qquad f(x)=\begin{cases}e^{-\frac{1}{x^{2}}}&x>0,\\ 0&x\leq 0.\end{cases}

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y i = { y i * if y i * > y L y L if y i * y L . subscript y i cases superscript subscript y i if superscript subscript y i subscript y L subscript y L if superscript subscript y i subscript y L y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{i}^{*}>y_{L}\\ y_{L}&\textrm{if}\;y_{i}^{*}\leq y_{L}.\end{cases}

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y 2 i = { y 2 i * if y 1 i * > 0 0 if y 1 i * 0. subscript y 2 i cases superscript subscript y 2 i if superscript subscript y 1 i 0 0 if superscript subscript y 1 i 0. y_{2i}=\begin{cases}y_{2i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}

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| a | = { a , if a ( p - 1 ) / 2 1 mod p , - a , if a ( p - 1 ) / 2 - 1 mod p . a cases a if superscript a p 1 2 modulo 1 p a if superscript a p 1 2 modulo 1 p \mathcal{|}a|=\begin{cases}a,&\textrm{if }a^{(p-1)/2}\equiv 1\bmod{p},\\ -a,&\textrm{if }a^{(p-1)/2}\equiv-1\bmod{p}.\end{cases}

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g = { - 5 9 ( x 2 + 1 ) S , if T > 1 2 5 9 ( x 2 + 1 ) S , if T < 1 2 0 , if T = 0 g cases normal-⋅ 5 9 superscript x 2 1 S if T 1 2 normal-⋅ 5 9 superscript x 2 1 S if T 1 2 0 if T 0 g=\begin{cases}-\sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T>\frac{1}{% 2}\\ \sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T<\frac{1}{2}\\ 0,&\mbox{if}~{}~{}T=0\\ \end{cases}

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I | X | K = { 1 if | X | K , 0 if | X | < K . subscript I X K cases 1 if X K 0 if X K I_{|X|\geq K}=\begin{cases}1&\,\text{if }|X|\geq K,\\ 0&\,\text{if }|X|<K.\end{cases}

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( I 1 I 2 ) = ( Y 11 Y 12 Y 21 Y 22 ) ( V 1 V 2 ) binomial subscript I 1 subscript I 2 subscript Y 11 subscript Y 12 subscript Y 21 subscript Y 22 binomial subscript V 1 subscript V 2 {I_{1}\choose I_{2}}=\begin{pmatrix}Y_{11}&Y_{12}\\ Y_{21}&Y_{22}\end{pmatrix}{V_{1}\choose V_{2}}

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[ 𝐒 ] = [ S 11 S 12 S 12 S 11 ] delimited-[] 𝐒 subscript S 11 subscript S 12 subscript S 12 subscript S 11 \left[\mathbf{S}\right]=\begin{bmatrix}S_{11}&S_{12}\\ S_{12}&S_{11}\end{bmatrix}

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[ F R 1 F R 2 ] = [ A 11 A 12 A 21 A 22 ] [ D P 1 D P 2 ] F subscript R 1 F subscript R 2 subscript A 11 subscript A 12 subscript A 21 subscript A 22 D subscript P 1 D subscript P 2 \begin{bmatrix}FR_{1}\\ FR_{2}\end{bmatrix}=\begin{bmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}DP_{1}\\ DP_{2}\end{bmatrix}

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𝐒 ( p ) = [ s 11 s 12 s 21 s 22 ] 𝐒 p subscript s 11 subscript s 12 subscript s 21 subscript s 22 \mathbf{S}(p)=\begin{bmatrix}s_{11}&s_{12}\\ s_{21}&s_{22}\end{bmatrix}

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𝐏 = [ 𝐏 11 𝐏 12 𝐏 21 𝐏 22 ] . 𝐏 subscript 𝐏 11 subscript 𝐏 12 subscript 𝐏 21 subscript 𝐏 22 \mathbf{P}=\begin{bmatrix}\mathbf{P}_{11}&\mathbf{P}_{12}\\ \mathbf{P}_{21}&\mathbf{P}_{22}\end{bmatrix}.

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| b 1 a 12 b 2 a 22 | = | a 11 x 1 a 12 a 21 x 1 a 22 | = x 1 | a 11 a 12 a 21 a 22 | subscript b 1 subscript a 12 subscript b 2 subscript a 22 subscript a 11 subscript x 1 subscript a 12 subscript a 21 subscript x 1 subscript a 22 subscript x 1 subscript a 11 subscript a 12 subscript a 21 subscript a 22 \begin{vmatrix}b_{1}&a_{12}\\ b_{2}&a_{22}\end{vmatrix}=\begin{vmatrix}a_{11}x_{1}&a_{12}\\ a_{21}x_{1}&a_{22}\end{vmatrix}=x_{1}\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}

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[ V 1 I 2 ] = [ 0 n - n 0 ] [ I 1 V 2 ] subscript V 1 subscript I 2 0 n n 0 subscript I 1 subscript V 2 \begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}0&n\\ -n&0\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}

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[ V 1 0 ] = [ R 1 + R 2 - R 2 - R 2 R 2 + R 3 ] [ I 1 I 2 ] subscript V 1 0 subscript R 1 subscript R 2 subscript R 2 subscript R 2 subscript R 2 subscript R 3 subscript I 1 subscript I 2 \begin{bmatrix}V_{1}\\ 0\end{bmatrix}=\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\ -R_{2}&R_{2}+R_{3}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}

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[ u v w ] = [ c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 ] [ a 1 a 2 a 3 ] u v w subscript c 11 subscript c 12 subscript c 13 subscript c 21 subscript c 22 subscript c 23 subscript c 31 subscript c 32 subscript c 33 subscript a 1 subscript a 2 subscript a 3 \begin{bmatrix}u\\ v\\ w\\ \end{bmatrix}=\begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}

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[ c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 ] [ x 1 x 2 x 3 ] = [ y 1 y 2 y 3 ] subscript c 11 subscript c 12 subscript c 13 subscript c 21 subscript c 22 subscript c 23 subscript c 31 subscript c 32 subscript c 33 subscript x 1 subscript x 2 subscript x 3 subscript y 1 subscript y 2 subscript y 3 \begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\end{bmatrix}

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[ V 1 I 2 ] = [ h 11 h 12 h 21 h 22 ] [ I 1 V 2 ] subscript V 1 subscript I 2 subscript h 11 subscript h 12 subscript h 21 subscript h 22 subscript I 1 subscript V 2 \begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}

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[ z v ] = 𝐏 ( s ) [ w u ] = [ P 11 ( s ) P 12 ( s ) P 21 ( s ) P 22 ( s ) ] [ w u ] z v 𝐏 s w u subscript P 11 s subscript P 12 s subscript P 21 s subscript P 22 s w u \begin{bmatrix}z\\ v\end{bmatrix}=\mathbf{P}(s)\,\begin{bmatrix}w\\ u\end{bmatrix}=\begin{bmatrix}P_{11}(s)&P_{12}(s)\\ P_{21}(s)&P_{22}(s)\end{bmatrix}\,\begin{bmatrix}w\\ u\end{bmatrix}

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H = [ H 11 H 12 H 12 H 22 ] H subscript H 11 subscript H 12 superscript subscript H 12 normal-∗ subscript H 22 H=\begin{bmatrix}H_{11}&H_{12}\\ H_{12}^{\ast}&H_{22}\end{bmatrix}

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[ v F ] = [ z 11 z 12 z 21 z 22 ] [ i u ] v F subscript z 11 subscript z 12 subscript z 21 subscript z 22 i u \begin{bmatrix}v\\ F\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}i\\ u\end{bmatrix}

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( V 1 V 2 ) = ( Z 11 Z 12 Z 21 Z 22 ) ( I 1 I 2 ) binomial subscript V 1 subscript V 2 subscript Z 11 subscript Z 12 subscript Z 21 subscript Z 22 binomial subscript I 1 subscript I 2 {V_{1}\choose V_{2}}=\begin{pmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{pmatrix}{I_{1}\choose I_{2}}

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[ f 1 g 1 ] = [ A B C D ] [ 0 y 0 ] subscript f 1 subscript g 1 A B C D 0 subscript y 0 \begin{bmatrix}f_{1}\\ g_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}0\\ y_{0}\end{bmatrix}

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T = [ T 11 T 12 0 T 22 ] T subscript T 11 subscript T 12 0 subscript T 22 T=\begin{bmatrix}T_{11}&T_{12}\\ 0&T_{22}\end{bmatrix}

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[ σ 11 σ 22 σ 12 ] = [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] [ ε 11 ε 22 ε 12 ] subscript σ 11 subscript σ 22 subscript σ 12 subscript C 11 subscript C 12 subscript C 13 subscript C 12 subscript C 22 subscript C 23 subscript C 13 subscript C 23 subscript C 33 subscript ε 11 subscript ε 22 subscript ε 12 \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}

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A Λ = [ 1 - 3 1 2 ] [ λ 1 λ 2 ] . A normal-Λ 1 3 1 2 subscript λ 1 subscript λ 2 A\Lambda=\begin{bmatrix}1&-3\\ 1&2\end{bmatrix}\begin{bmatrix}\lambda_{1}\\ \lambda_{2}\end{bmatrix}.\,\!

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[ 4 3 6 3 ] = [ l 11 0 l 21 l 22 ] [ u 11 u 12 0 u 22 ] . 4 3 6 3 subscript l 11 0 subscript l 21 subscript l 22 subscript u 11 subscript u 12 0 subscript u 22 \begin{bmatrix}4&3\\ 6&3\end{bmatrix}=\begin{bmatrix}l_{11}&0\\ l_{21}&l_{22}\end{bmatrix}\begin{bmatrix}u_{11}&u_{12}\\ 0&u_{22}\end{bmatrix}.

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[ Z 11 Z 12 Z 21 Z 22 ] = [ n 11 n 12 n 21 n 22 ] [ Z 11 Z 12 Z 21 Z 22 ] [ n 11 n 12 n 21 n 22 ] subscript superscript Z normal-′ 11 subscript superscript Z normal-′ 12 subscript superscript Z normal-′ 21 subscript superscript Z normal-′ 22 subscript n 11 subscript n 12 subscript n 21 subscript n 22 subscript Z 11 subscript Z 12 subscript Z 21 subscript Z 22 subscript n 11 subscript n 12 subscript n 21 subscript n 22 \begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}n_{11}&n_{12}\\ n_{21}&n_{22}\end{bmatrix}\begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix}\begin{bmatrix}n_{11}&n_{12}\\ n_{21}&n_{22}\end{bmatrix}

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[ Z 11 Z 12 Z 21 Z 22 ] = [ Z 11 Z 12 Z 21 Z 22 ] - 1 subscript superscript Z normal-′ 11 subscript superscript Z normal-′ 12 subscript superscript Z normal-′ 21 subscript superscript Z normal-′ 22 superscript subscript Z 11 subscript Z 12 subscript Z 21 subscript Z 22 1 \begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix}^{-1}

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𝐂 = ( 𝐂 11 𝐂 12 𝐂 21 𝐂 22 ) = ( 𝐀 11 𝐀 12 𝐀 21 𝐀 22 ) ( 𝐁 11 𝐁 12 𝐁 21 𝐁 22 ) = 𝐀𝐁 𝐂 subscript 𝐂 11 subscript 𝐂 12 subscript 𝐂 21 subscript 𝐂 22 subscript 𝐀 11 subscript 𝐀 12 subscript 𝐀 21 subscript 𝐀 22 subscript 𝐁 11 subscript 𝐁 12 subscript 𝐁 21 subscript 𝐁 22 𝐀𝐁 \mathbf{C}=\begin{pmatrix}\mathbf{C}_{11}&\mathbf{C}_{12}\\ \mathbf{C}_{21}&\mathbf{C}_{22}\\ \end{pmatrix}=\begin{pmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\\ \end{pmatrix}\begin{pmatrix}\mathbf{B}_{11}&\mathbf{B}_{12}\\ \mathbf{B}_{21}&\mathbf{B}_{22}\\ \end{pmatrix}=\mathbf{A}\mathbf{B}

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[ V F ] = [ z 11 z 12 z 21 z 22 ] [ I v ] V F subscript z 11 subscript z 12 subscript z 21 subscript z 22 I v \begin{bmatrix}V\\ F\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I\\ v\end{bmatrix}

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[ i u ] = [ y 11 y 12 y 21 y 22 ] [ v F ] i u subscript y 11 subscript y 12 subscript y 21 subscript y 22 v F \begin{bmatrix}i\\ u\end{bmatrix}=\begin{bmatrix}y_{11}&y_{12}\\ y_{21}&y_{22}\end{bmatrix}\begin{bmatrix}v\\ F\end{bmatrix}

NTCIR12-MathWiki-9rate: 1

[ V 1 V 0 ] = [ z ( j ω ) 11 z ( j ω ) 12 z ( j ω ) 21 z ( j ω ) 22 ] [ I 1 I 0 ] subscript V 1 subscript V 0 z subscript j ω 11 z subscript j ω 12 z subscript j ω 21 z subscript j ω 22 subscript I 1 subscript I 0 \begin{bmatrix}V_{1}\\ V_{0}\end{bmatrix}=\begin{bmatrix}z(j\omega)_{11}&z(j\omega)_{12}\\ z(j\omega)_{21}&z(j\omega)_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{0}\end{bmatrix}

NTCIR12-MathWiki-9rate: 0

M = ( M 11 M 12 M 21 M 22 ) M subscript M 11 subscript M 12 subscript M 21 subscript M 22 M=\left(\begin{matrix}{{M}_{11}}&{{M}_{12}}\\ {{M}_{21}}&{{M}_{22}}\\ \end{matrix}\right)

NTCIR12-MathWiki-9rate: 1

[ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] [ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] subscript σ 1 subscript σ 2 subscript σ 3 subscript σ 4 subscript σ 5 subscript σ 6 subscript C 11 subscript C 12 subscript C 13 subscript C 14 subscript C 15 subscript C 16 subscript C 12 subscript C 22 subscript C 23 subscript C 24 subscript C 25 subscript C 26 subscript C 13 subscript C 23 subscript C 33 subscript C 34 subscript C 35 subscript C 36 subscript C 14 subscript C 24 subscript C 34 subscript C 44 subscript C 45 subscript C 46 subscript C 15 subscript C 25 subscript C 35 subscript C 45 subscript C 55 subscript C 56 subscript C 16 subscript C 26 subscript C 36 subscript C 46 subscript C 56 subscript C 66 subscript ε 1 subscript ε 2 subscript ε 3 subscript ε 4 subscript ε 5 subscript ε 6 \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{1% 6}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}\begin{bmatrix}% \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \varepsilon_{6}\end{bmatrix}

NTCIR12-MathWiki-9rate: 1

[ σ 11 σ 22 σ 12 ] = [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] [ ε 11 ε 22 ε 12 ] subscript σ 11 subscript σ 22 subscript σ 12 subscript C 11 subscript C 12 subscript C 13 subscript C 12 subscript C 22 subscript C 23 subscript C 13 subscript C 23 subscript C 33 subscript ε 11 subscript ε 22 subscript ε 12 \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}

NTCIR12-MathWiki-9rate: 3

[ V 1 V 2 ] = [ z 11 z 12 z 21 z 22 ] [ I 1 I 2 ] subscript V 1 subscript V 2 subscript z 11 subscript z 12 subscript z 21 subscript z 22 subscript I 1 subscript I 2 \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}

NTCIR12-MathWiki-9rate: 0

[ K 11 K 12 K 21 K 22 ] [ x 1 x 2 ] = [ F 1 F 2 ] subscript K 11 subscript K 12 subscript K 21 subscript K 22 subscript x 1 subscript x 2 subscript F 1 subscript F 2 \begin{bmatrix}K_{11}&K_{12}\\ K_{21}&K_{22}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}=\begin{bmatrix}F_{1}\\ F_{2}\end{bmatrix}

NTCIR12-MathWiki-9rate: 0

( B C ) = ( S 11 S 12 S 21 S 22 ) ( A D ) . B C subscript S 11 subscript S 12 subscript S 21 subscript S 22 A D \begin{pmatrix}B\\ C\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}A\\ D\end{pmatrix}.

NTCIR12-MathWiki-9rate: 1

( b 1 a 1 ) = ( T 11 T 12 T 21 T 22 ) ( a 2 b 2 ) subscript b 1 subscript a 1 subscript T 11 subscript T 12 subscript T 21 subscript T 22 subscript a 2 subscript b 2 \begin{pmatrix}b_{1}\\ a_{1}\end{pmatrix}=\begin{pmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{pmatrix}\begin{pmatrix}a_{2}\\ b_{2}\end{pmatrix}\,

NTCIR12-MathWiki-9rate: 1

( b 1 b 2 ) = ( S 11 S 12 S 21 S 22 ) ( a 1 a 2 ) subscript b 1 subscript b 2 subscript S 11 subscript S 12 subscript S 21 subscript S 22 subscript a 1 subscript a 2 \begin{pmatrix}b_{1}\\ b_{2}\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}a_{1}\\ a_{2}\end{pmatrix}\,

NTCIR12-MathWiki-9rate: 1

[ x 1 x 2 * ] = [ h 1 - h 2 h 2 * h 1 * ] [ S 1 S 2 * ] + [ n 1 n 2 * ] subscript x 1 superscript subscript x 2 subscript h 1 subscript h 2 superscript subscript h 2 superscript subscript h 1 subscript S 1 superscript subscript S 2 subscript n 1 superscript subscript n 2 \begin{bmatrix}x_{1}\\ x_{2}^{*}\end{bmatrix}=\begin{bmatrix}h_{1}&-h_{2}\\ h_{2}^{*}&h_{1}^{*}\end{bmatrix}\begin{bmatrix}S_{1}\\ S_{2}^{*}\end{bmatrix}+\begin{bmatrix}n_{1}\\ n_{2}^{*}\end{bmatrix}

NTCIR12-MathWiki-9rate: 0

[ T 1 T 2 T 3 ] = [ σ 11 σ 21 σ 31 σ 12 σ 22 σ 32 σ 13 σ 23 σ 33 ] [ n 1 n 2 n 3 ] subscript T 1 subscript T 2 subscript T 3 subscript σ 11 subscript σ 21 subscript σ 31 subscript σ 12 subscript σ 22 subscript σ 32 subscript σ 13 subscript σ 23 subscript σ 33 subscript n 1 subscript n 2 subscript n 3 \begin{bmatrix}T_{1}\\ T_{2}\\ T_{3}\end{bmatrix}=\begin{bmatrix}\sigma_{11}&\sigma_{21}&\sigma_{31}\\ \sigma_{12}&\sigma_{22}&\sigma_{32}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{bmatrix}\begin{bmatrix}n_{1}\\ n_{2}\\ n_{3}\end{bmatrix}

NTCIR12-MathWiki-9rate: 0

[ V 1 V 2 ] = - t / 2 t / 2 [ σ 13 σ 23 ] d x 3 . subscript V 1 subscript V 2 superscript subscript t 2 t 2 subscript σ 13 subscript σ 23 d subscript x 3 \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\int_{-t/2}^{t/2}\begin{bmatrix}\sigma_{13}\\ \sigma_{23}\end{bmatrix}\,dx_{3}\,.

NTCIR12-MathWiki-9rate: 0

G = ( G 11 G 12 G 21 G 22 ) G subscript G 11 subscript G 12 subscript G 21 subscript G 22 G=\begin{pmatrix}G_{11}&G_{12}\\ G_{21}&G_{22}\end{pmatrix}

NTCIR12-MathWiki-9rate: 0

[ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] [ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] subscript σ 1 subscript σ 2 subscript σ 3 subscript σ 4 subscript σ 5 subscript σ 6 subscript C 11 subscript C 12 subscript C 13 subscript C 14 subscript C 15 subscript C 16 subscript C 12 subscript C 22 subscript C 23 subscript C 24 subscript C 25 subscript C 26 subscript C 13 subscript C 23 subscript C 33 subscript C 34 subscript C 35 subscript C 36 subscript C 14 subscript C 24 subscript C 34 subscript C 44 subscript C 45 subscript C 46 subscript C 15 subscript C 25 subscript C 35 subscript C 45 subscript C 55 subscript C 56 subscript C 16 subscript C 26 subscript C 36 subscript C 46 subscript C 56 subscript C 66 subscript ε 1 subscript ε 2 subscript ε 3 subscript ε 4 subscript ε 5 subscript ε 6 \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{1% 6}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}\begin{bmatrix}% \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \varepsilon_{6}\end{bmatrix}

NTCIR12-MathWiki-9rate: 4

[ V 1 I 2 ] = [ h 11 h 12 h 21 h 22 ] [ I 1 V 2 ] subscript V 1 subscript I 2 subscript h 11 subscript h 12 subscript h 21 subscript h 22 subscript I 1 subscript V 2 \begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}

NTCIR12-MathWiki-9rate: 2

[ b 1 b 2 ] = [ S 11 S 12 S 21 S 22 ] [ a 1 a 2 ] subscript b 1 subscript b 2 subscript S 11 subscript S 12 subscript S 21 subscript S 22 subscript a 1 subscript a 2 \begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}=\begin{bmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\end{bmatrix}

NTCIR12-MathWiki-9rate: 0

[ y 1 , t y 2 , t ] = [ c 1 c 2 ] + [ A 1 , 1 A 1 , 2 A 2 , 1 A 2 , 2 ] [ y 1 , t - 1 y 2 , t - 1 ] + [ e 1 , t e 2 , t ] , subscript y 1 t subscript y 2 t subscript c 1 subscript c 2 subscript A 1 1 subscript A 1 2 subscript A 2 1 subscript A 2 2 subscript y 1 t 1 subscript y 2 t 1 subscript e 1 t subscript e 2 t \begin{bmatrix}y_{1,t}\\ y_{2,t}\end{bmatrix}=\begin{bmatrix}c_{1}\\ c_{2}\end{bmatrix}+\begin{bmatrix}A_{1,1}&A_{1,2}\\ A_{2,1}&A_{2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1}\\ y_{2,t-1}\end{bmatrix}+\begin{bmatrix}e_{1,t}\\ e_{2,t}\end{bmatrix},